{ "schema_version": 1, "generated_from": { "claudiverse_commit": "4a88328", "generated_at": "2026-10-10T11:13:12Z" }, "node_types": [ "book", "person", "chapter", "exercise_set", "problem", "problem_form", "problem_shape", "concept", "method", "theorem", "law", "quantity", "unit", "instrument", "experiment", "excerpt", "equation", "capability" ], "edge_types": [ "written_by", "part_of", "taught_in", "practices", "quoted_from", "explains", "appears_in", "states", "relates", "instance_of", "needs", "prerequisite_of", "special_case_of", "generalizes", "uses", "inverse_of", "contrasts_with", "measures", "unit_of", "named_after", "discovered_by", "related_to" ], "book": { "id": "dickson-theory-of-equations-1922", "title": "First Course in the Theory of Equations", "authors": [ "Leonard Eugene Dickson" ], "year": 1922, "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "transcription": { "source": "Project Gutenberg eBook #29785", "url": "https://www.gutenberg.org/ebooks/29785", "released": "August 25, 2009", "licence_terms": "This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org" }, "file": "books/dickson-theory-of-equations-1922.json" }, "chapters": [ { "id": "dickson-theory-of-equations-1922/ch-i", "number": "I", "title": "Complex Numbers", "name": "Dickson 1922, ch. I: Complex Numbers", "pages": [ "1", "11" ], "concepts": [], "excerpts": [], "equations": [ "dickson-theory-of-equations-1922/eq-498dcea9b8", "dickson-theory-of-equations-1922/eq-ae3fd510eb", "dickson-theory-of-equations-1922/eq-a091be94e3", "dickson-theory-of-equations-1922/eq-0961b35b5e", "dickson-theory-of-equations-1922/eq-0756b518ce", "dickson-theory-of-equations-1922/eq-7d97384470", "dickson-theory-of-equations-1922/eq-728c6eae67", "dickson-theory-of-equations-1922/eq-4338bef33c", "dickson-theory-of-equations-1922/eq-c3515f4ce7", "dickson-theory-of-equations-1922/eq-7932989444", "dickson-theory-of-equations-1922/eq-d42a1beb66", "dickson-theory-of-equations-1922/eq-98a5aaf591", "dickson-theory-of-equations-1922/eq-238d308122", "dickson-theory-of-equations-1922/eq-f6de462048", "dickson-theory-of-equations-1922/eq-bf5c9ba49e", "dickson-theory-of-equations-1922/eq-c0a9d3cd5a", "dickson-theory-of-equations-1922/eq-5e9b19b016", "dickson-theory-of-equations-1922/eq-dbefbba7cb", "dickson-theory-of-equations-1922/eq-c5a6b9f2ee", "dickson-theory-of-equations-1922/eq-f7cdff540f", "dickson-theory-of-equations-1922/eq-8140068f41", "dickson-theory-of-equations-1922/eq-b19df795f5", "dickson-theory-of-equations-1922/eq-7887749d4b", "dickson-theory-of-equations-1922/eq-2fa905e224", "dickson-theory-of-equations-1922/eq-4e493c3b1b", "dickson-theory-of-equations-1922/eq-9ac58b7b2a", "dickson-theory-of-equations-1922/eq-92e15a3d1c", "dickson-theory-of-equations-1922/eq-53ce9296a1", "dickson-theory-of-equations-1922/eq-8f53e62748", "dickson-theory-of-equations-1922/eq-aaea467059", "dickson-theory-of-equations-1922/eq-b88d06a9bb", "dickson-theory-of-equations-1922/eq-09d00b99d9", "dickson-theory-of-equations-1922/eq-db6f90e194", "dickson-theory-of-equations-1922/eq-74d3968fc7", "dickson-theory-of-equations-1922/eq-4c66e7b31b", "dickson-theory-of-equations-1922/eq-57c8d2fa81", "dickson-theory-of-equations-1922/eq-504c7a4acc", "dickson-theory-of-equations-1922/eq-cf992cb854", "dickson-theory-of-equations-1922/eq-82fe1399d0" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page2", "dickson-theory-of-equations-1922/ex-page6", "dickson-theory-of-equations-1922/ex-page9", "dickson-theory-of-equations-1922/ex-page10" ] }, { "id": "dickson-theory-of-equations-1922/ch-ii", "number": "II", "title": "Elementary Theorems on the Roots of an Equation", "name": "Dickson 1922, ch. II: Elementary Theorems on the Roots of an Equation", "pages": [ "24", "29" ], "concepts": [ "concept/complex-number", "concept/discriminant", "concept/factored-form", "concept/linear-factor", "concept/lower-limit-to-the-roots", "concept/multiple-root", "concept/root-of-an-equation", "concept/upper-limit-to-the-roots", "method/mathematical-induction", "method/quadratic-formula", "method/synthetic-division", "theorem/equation-of-degree-n-has-at-most-n-roots", "theorem/factor-theorem", "theorem/fundamental-theorem-of-algebra", "theorem/imaginary-roots-occur-in-pairs", "theorem/relations-between-roots-and-coefficients", "theorem/remainder-theorem", "theorem/upper-limit-theorem-by-coefficient-ratios", "theorem/upper-limit-theorem-by-greatest-negative-coefficient" ], "excerpts": [ "dickson-theory-of-equations-1922/x-a05f5e1381", "dickson-theory-of-equations-1922/x-8ad9ce7628", "dickson-theory-of-equations-1922/x-e00a7f71e7", "dickson-theory-of-equations-1922/x-25f2cde19e", "dickson-theory-of-equations-1922/x-b35ac51792", "dickson-theory-of-equations-1922/x-4fe43fe1cf", "dickson-theory-of-equations-1922/x-9f8bfb4329" ], "equations": [ "dickson-theory-of-equations-1922/eq-7d302667ca", "dickson-theory-of-equations-1922/eq-82980b7e36", "dickson-theory-of-equations-1922/eq-959a6e3ba9", "dickson-theory-of-equations-1922/eq-42f482698a", "dickson-theory-of-equations-1922/eq-65ac0615df", "dickson-theory-of-equations-1922/eq-49606eed28", "dickson-theory-of-equations-1922/eq-81959ac762", "dickson-theory-of-equations-1922/eq-d535ca0ab0", "dickson-theory-of-equations-1922/eq-3d686061ea", "dickson-theory-of-equations-1922/eq-24c3dbbbf9", "dickson-theory-of-equations-1922/eq-3dc51bfdb1", "dickson-theory-of-equations-1922/eq-0828096f6e", "dickson-theory-of-equations-1922/eq-62542de319", "dickson-theory-of-equations-1922/eq-d92a6b31ec", "dickson-theory-of-equations-1922/eq-2f0fc55a5e", "dickson-theory-of-equations-1922/eq-c492f00446", "dickson-theory-of-equations-1922/eq-6b6d9c3e1c", "dickson-theory-of-equations-1922/eq-68ed424c59", "dickson-theory-of-equations-1922/eq-52bbe6e89b", "dickson-theory-of-equations-1922/eq-f1e0fc6d07", "dickson-theory-of-equations-1922/eq-d1f8cc0481", "dickson-theory-of-equations-1922/eq-e1a8a45424", "dickson-theory-of-equations-1922/eq-841c5f4bce", "dickson-theory-of-equations-1922/eq-72621e8045", "dickson-theory-of-equations-1922/eq-5c36b0cdd4", "dickson-theory-of-equations-1922/eq-61b7702517", "dickson-theory-of-equations-1922/eq-1a457adaa8", "dickson-theory-of-equations-1922/eq-ed0dc2a88e", "dickson-theory-of-equations-1922/eq-fcd915bb34", "dickson-theory-of-equations-1922/eq-63afa8a92e", "dickson-theory-of-equations-1922/eq-c463c3d0b2", "dickson-theory-of-equations-1922/eq-50dc27ba42", "dickson-theory-of-equations-1922/eq-3141860cb9", "dickson-theory-of-equations-1922/eq-b7dd6bc71a", "dickson-theory-of-equations-1922/eq-2896e2d772", "dickson-theory-of-equations-1922/eq-744aeec469", "dickson-theory-of-equations-1922/eq-f90c60584a", "dickson-theory-of-equations-1922/eq-175d468f2c", "dickson-theory-of-equations-1922/eq-fdeb715f60" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page25", "dickson-theory-of-equations-1922/ex-page26", "dickson-theory-of-equations-1922/ex-page27", "dickson-theory-of-equations-1922/ex-page28", "dickson-theory-of-equations-1922/ex-page13", "dickson-theory-of-equations-1922/ex-page15", "dickson-theory-of-equations-1922/ex-page17", "dickson-theory-of-equations-1922/ex-page19", "dickson-theory-of-equations-1922/ex-page20", "dickson-theory-of-equations-1922/ex-page23" ] }, { "id": "dickson-theory-of-equations-1922/ch-iii", "number": "III", "title": "Constructions with Ruler and Compasses", "name": "Dickson 1922, ch. III: Constructions with Ruler and Compasses", "pages": [ "29", "45" ], "concepts": [], "excerpts": [], "equations": [ "dickson-theory-of-equations-1922/eq-7ec82acf8c", "dickson-theory-of-equations-1922/eq-051f7a6f18", "dickson-theory-of-equations-1922/eq-c0b1cf9c5b", "dickson-theory-of-equations-1922/eq-6fac5553c3", "dickson-theory-of-equations-1922/eq-eb38515661", "dickson-theory-of-equations-1922/eq-7ec99d9584", "dickson-theory-of-equations-1922/eq-fcfbab8685", "dickson-theory-of-equations-1922/eq-1042fbdf3f", "dickson-theory-of-equations-1922/eq-3734d44d89", "dickson-theory-of-equations-1922/eq-7035756e71", "dickson-theory-of-equations-1922/eq-ec9001ece3", "dickson-theory-of-equations-1922/eq-cd2d133997", "dickson-theory-of-equations-1922/eq-b508460d24", "dickson-theory-of-equations-1922/eq-a977661c4a", "dickson-theory-of-equations-1922/eq-2b55e8aed3", "dickson-theory-of-equations-1922/eq-4ee77a2fcc", "dickson-theory-of-equations-1922/eq-26c685639e", "dickson-theory-of-equations-1922/eq-6492bf3ef7", "dickson-theory-of-equations-1922/eq-b79aff2c45", "dickson-theory-of-equations-1922/eq-c220d80036", "dickson-theory-of-equations-1922/eq-5e538f39c6", "dickson-theory-of-equations-1922/eq-67e891a912", "dickson-theory-of-equations-1922/eq-5c388569d0", "dickson-theory-of-equations-1922/eq-b01e00317d", "dickson-theory-of-equations-1922/eq-70f9c72cd5", "dickson-theory-of-equations-1922/eq-e329073739", "dickson-theory-of-equations-1922/eq-c5827ece03", "dickson-theory-of-equations-1922/eq-5481ae5491", "dickson-theory-of-equations-1922/eq-ff10398641", "dickson-theory-of-equations-1922/eq-7ded9bbf7a", "dickson-theory-of-equations-1922/eq-012ec9707c", "dickson-theory-of-equations-1922/eq-48711b4d4d", "dickson-theory-of-equations-1922/eq-d4c872d09a", "dickson-theory-of-equations-1922/eq-a64f147991", "dickson-theory-of-equations-1922/eq-afbce9cf36", "dickson-theory-of-equations-1922/eq-18ac9e662d", "dickson-theory-of-equations-1922/eq-7556512d7c", "dickson-theory-of-equations-1922/eq-afd1a67805", "dickson-theory-of-equations-1922/eq-69b49ea4e1", "dickson-theory-of-equations-1922/eq-56eefb0d98", "dickson-theory-of-equations-1922/eq-0b11f3e3fa", "dickson-theory-of-equations-1922/eq-2691c26fd1", "dickson-theory-of-equations-1922/eq-dd15919ba8", "dickson-theory-of-equations-1922/eq-036d6e2de2", "dickson-theory-of-equations-1922/eq-a8012a68aa", "dickson-theory-of-equations-1922/eq-776c7f347f", "dickson-theory-of-equations-1922/eq-79adf0e5ff", "dickson-theory-of-equations-1922/eq-bf244731b0", "dickson-theory-of-equations-1922/eq-4d3eaab8e1", "dickson-theory-of-equations-1922/eq-9b36a1ffcb", "dickson-theory-of-equations-1922/eq-d27c7b37fb", "dickson-theory-of-equations-1922/eq-739a438991", "dickson-theory-of-equations-1922/eq-147b3b4862", "dickson-theory-of-equations-1922/eq-c151e5de6f", "dickson-theory-of-equations-1922/eq-3db1099072", "dickson-theory-of-equations-1922/eq-a5b3b0f054", "dickson-theory-of-equations-1922/eq-2fb1f035ce", "dickson-theory-of-equations-1922/eq-11b5480f09", "dickson-theory-of-equations-1922/eq-31bfd5a67c", "dickson-theory-of-equations-1922/eq-b7e47d7070", "dickson-theory-of-equations-1922/eq-a46a7513f5", "dickson-theory-of-equations-1922/eq-aa219b353a", "dickson-theory-of-equations-1922/eq-ca6a3c5cbf", "dickson-theory-of-equations-1922/eq-a09c5c1387", "dickson-theory-of-equations-1922/eq-f3ffcaf514", "dickson-theory-of-equations-1922/eq-412c876665", "dickson-theory-of-equations-1922/eq-17846b7dfe", "dickson-theory-of-equations-1922/eq-a1dcae5abc", "dickson-theory-of-equations-1922/eq-483c5e4f1c", "dickson-theory-of-equations-1922/eq-dddb132073", "dickson-theory-of-equations-1922/eq-dfc34646c5", "dickson-theory-of-equations-1922/eq-d470d87629", "dickson-theory-of-equations-1922/eq-de0ee14927", "dickson-theory-of-equations-1922/eq-28d7fcac2f", "dickson-theory-of-equations-1922/eq-87a089ce8d", "dickson-theory-of-equations-1922/eq-5f1212c0c5", "dickson-theory-of-equations-1922/eq-57a1d6f8a9", "dickson-theory-of-equations-1922/eq-31020f0b49", "dickson-theory-of-equations-1922/eq-2e0afa26fe", "dickson-theory-of-equations-1922/eq-9236ec48cf", "dickson-theory-of-equations-1922/eq-55a3a70e08", "dickson-theory-of-equations-1922/eq-c2b1e0e29b", "dickson-theory-of-equations-1922/eq-9d40885121", "dickson-theory-of-equations-1922/eq-812d01f5d5" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page30", "dickson-theory-of-equations-1922/ex-page40", "dickson-theory-of-equations-1922/ex-page44" ] }, { "id": "dickson-theory-of-equations-1922/ch-iv", "number": "IV", "title": "Solution of Cubic and Quartic Equations; Their Discriminants", "name": "Dickson 1922, ch. IV: Solution of Cubic and Quartic Equations; Their Discriminants", "pages": [ "45", "55" ], "concepts": [ "concept/cube-root-of-unity", "concept/cubic-equation", "concept/discriminant", "concept/irreducible-case", "concept/quartic-equation", "concept/reduced-cubic-equation", "concept/reduced-quartic-equation", "concept/resolvent-cubic", "method/descartes-solution-of-the-quartic-equation", "method/ferrari-s-solution-of-the-quartic-equation", "method/trigonometric-solution-of-a-cubic", "person/girolamo-cardano", "person/lodovico-ferrari", "person/ren-descartes", "theorem/cardan-s-formulas", "theorem/discriminant-formula-for-the-cubic", "theorem/discriminant-of-a-quartic-equals-that-of-its-resolvent-cubic", "theorem/number-of-real-roots-of-a-cubic" ], "excerpts": [ "dickson-theory-of-equations-1922/x-e28b954deb", "dickson-theory-of-equations-1922/x-9c839e64b5", "dickson-theory-of-equations-1922/x-f8e49eaff1", "dickson-theory-of-equations-1922/x-b37e967bc0", "dickson-theory-of-equations-1922/x-234ee0c0f6", "dickson-theory-of-equations-1922/x-e5ac8e5fc0", "dickson-theory-of-equations-1922/x-ce027671f0", "dickson-theory-of-equations-1922/x-d711aab631" ], "equations": [ "dickson-theory-of-equations-1922/eq-e7621149f0", "dickson-theory-of-equations-1922/eq-3c8c1052e2", "dickson-theory-of-equations-1922/eq-6596361c2f", "dickson-theory-of-equations-1922/eq-4e83469fb5", "dickson-theory-of-equations-1922/eq-98cba7d65d", "dickson-theory-of-equations-1922/eq-7b5584235b", "dickson-theory-of-equations-1922/eq-09f265c6d4", "dickson-theory-of-equations-1922/eq-6925d38e51", "dickson-theory-of-equations-1922/eq-99077da007", "dickson-theory-of-equations-1922/eq-2de7d35a68", "dickson-theory-of-equations-1922/eq-838f091d7a", "dickson-theory-of-equations-1922/eq-5e33576228", "dickson-theory-of-equations-1922/eq-137814bd93", "dickson-theory-of-equations-1922/eq-a2d7477e1e", "dickson-theory-of-equations-1922/eq-e057477cfa", "dickson-theory-of-equations-1922/eq-2cb1eaaa80", "dickson-theory-of-equations-1922/eq-129f5917fd", "dickson-theory-of-equations-1922/eq-8583a9fea2", "dickson-theory-of-equations-1922/eq-5fc07deaa2", "dickson-theory-of-equations-1922/eq-9d97af5edc", "dickson-theory-of-equations-1922/eq-77315f5426", "dickson-theory-of-equations-1922/eq-74c69d1aeb", "dickson-theory-of-equations-1922/eq-6471c37b12", "dickson-theory-of-equations-1922/eq-f0881680b8", "dickson-theory-of-equations-1922/eq-8034c0ec51", "dickson-theory-of-equations-1922/eq-db2a962045", "dickson-theory-of-equations-1922/eq-6a497c2f9d", "dickson-theory-of-equations-1922/eq-91be38c8e2", "dickson-theory-of-equations-1922/eq-265ee3ae11", "dickson-theory-of-equations-1922/eq-5aaac54a36", "dickson-theory-of-equations-1922/eq-00e50a8546", "dickson-theory-of-equations-1922/eq-7764c3a6b3", "dickson-theory-of-equations-1922/eq-a11aa4e10c", "dickson-theory-of-equations-1922/eq-93454282cb", "dickson-theory-of-equations-1922/eq-1cfa74d7d1", "dickson-theory-of-equations-1922/eq-44bd77719d", "dickson-theory-of-equations-1922/eq-4e6014cb4b", "dickson-theory-of-equations-1922/eq-736c0060d8", "dickson-theory-of-equations-1922/eq-3ce9cec79a", "dickson-theory-of-equations-1922/eq-ee84fe1e31", "dickson-theory-of-equations-1922/eq-7b4fcb73f0", "dickson-theory-of-equations-1922/eq-2174b10c59", "dickson-theory-of-equations-1922/eq-ad4801ebf6", "dickson-theory-of-equations-1922/eq-1f55d0cb8b", "dickson-theory-of-equations-1922/eq-21c1b15cff", "dickson-theory-of-equations-1922/eq-c7221081c9", "dickson-theory-of-equations-1922/eq-cac9022cf1", "dickson-theory-of-equations-1922/eq-439a41945c", "dickson-theory-of-equations-1922/eq-91019bd7ed", "dickson-theory-of-equations-1922/eq-dab3b0efc5", "dickson-theory-of-equations-1922/eq-bf5a941106", "dickson-theory-of-equations-1922/eq-c008449efe", "dickson-theory-of-equations-1922/eq-52fe7e096e", "dickson-theory-of-equations-1922/eq-412d4651ba", "dickson-theory-of-equations-1922/eq-42f45803e8", "dickson-theory-of-equations-1922/eq-61a93dd16a", "dickson-theory-of-equations-1922/eq-9d740d680a", "dickson-theory-of-equations-1922/eq-786adabbfb", "dickson-theory-of-equations-1922/eq-cd6c867298", "dickson-theory-of-equations-1922/eq-312e8df0ad", "dickson-theory-of-equations-1922/eq-5b68683450", "dickson-theory-of-equations-1922/eq-40e6fa3a96", "dickson-theory-of-equations-1922/eq-6b7c2caf8a" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page46", "dickson-theory-of-equations-1922/ex-page48", "dickson-theory-of-equations-1922/ex-page49", "dickson-theory-of-equations-1922/ex-page49b", "dickson-theory-of-equations-1922/ex-page51", "dickson-theory-of-equations-1922/ex-page52", "dickson-theory-of-equations-1922/ex-page53", "dickson-theory-of-equations-1922/ex-page54" ] }, { "id": "dickson-theory-of-equations-1922/ch-v", "number": "V", "title": "The Graph of an Equation", "name": "Dickson 1922, ch. V: The Graph of an Equation", "pages": [ "55", "71" ], "concepts": [ "concept/abscissa", "concept/bend-point", "concept/conjugate-complex-numbers", "concept/cubic-equation", "concept/derivative", "concept/factorial", "concept/graph-of-a-function", "concept/higher-order-derivative", "concept/highest-common-factor", "concept/imaginary-root", "concept/infinity", "concept/inflection-point", "concept/inflection-tangent", "concept/maximum", "concept/minimum", "concept/multiple-root", "concept/ordinary-tangent", "concept/ordinate", "concept/origin", "concept/point-of-tangency", "concept/polynomial", "concept/quadratic-equation", "concept/real-root", "concept/reduced-cubic-equation", "concept/root-of-an-equation", "concept/sign-of-a-polynomial-at-infinity", "concept/slope-of-a-curve", "concept/tangent", "method/differentiation", "method/graphical-solution-of-an-equation", "method/plotting-a-curve", "method/solving-an-equation-graphically", "theorem/intermediate-value-theorem", "theorem/power-rule-for-differentiation", "theorem/rolle-s-theorem", "theorem/sum-rule-for-differentiation", "theorem/taylor-s-theorem" ], "excerpts": [ "dickson-theory-of-equations-1922/x-41a01ff24a", "dickson-theory-of-equations-1922/x-a93a96ffb4", "dickson-theory-of-equations-1922/x-aaaf2f808e", "dickson-theory-of-equations-1922/x-7e0fc26cea", "dickson-theory-of-equations-1922/x-6e76e36cf4", "dickson-theory-of-equations-1922/x-6830ce5423", "dickson-theory-of-equations-1922/x-fb53dcb8e9", "dickson-theory-of-equations-1922/x-d3c80f6dcb", "dickson-theory-of-equations-1922/x-b64fc54d3f", "dickson-theory-of-equations-1922/x-adb0691fc0", "dickson-theory-of-equations-1922/x-ea7d4d2020", "dickson-theory-of-equations-1922/x-3bf441d44c" ], "equations": [ "dickson-theory-of-equations-1922/eq-db5d67deec", "dickson-theory-of-equations-1922/eq-6b933c787c", "dickson-theory-of-equations-1922/eq-1901272b31", "dickson-theory-of-equations-1922/eq-6758a72b9a", "dickson-theory-of-equations-1922/eq-fdf49d6e1c", "dickson-theory-of-equations-1922/eq-f8f5dc6e67", "dickson-theory-of-equations-1922/eq-c409af3e74", "dickson-theory-of-equations-1922/eq-21a5ab2501", "dickson-theory-of-equations-1922/eq-693f81b1fb", "dickson-theory-of-equations-1922/eq-966d1ba68d", "dickson-theory-of-equations-1922/eq-647b90521e", "dickson-theory-of-equations-1922/eq-8dd67f90f9", "dickson-theory-of-equations-1922/eq-2961943cc5", "dickson-theory-of-equations-1922/eq-1ec24a4431", "dickson-theory-of-equations-1922/eq-0244af2213", "dickson-theory-of-equations-1922/eq-ad9fce4e94", "dickson-theory-of-equations-1922/eq-7d5977b7db", "dickson-theory-of-equations-1922/eq-39c6da23aa", "dickson-theory-of-equations-1922/eq-32a16d1e96", "dickson-theory-of-equations-1922/eq-d30219e428", "dickson-theory-of-equations-1922/eq-8b024973a0", "dickson-theory-of-equations-1922/eq-25e92bf677", "dickson-theory-of-equations-1922/eq-166d10edce", "dickson-theory-of-equations-1922/eq-2f7718e15f", "dickson-theory-of-equations-1922/eq-db7be62b8a", "dickson-theory-of-equations-1922/eq-cacc5c228e", "dickson-theory-of-equations-1922/eq-7c20e94c81" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page55", "dickson-theory-of-equations-1922/ex-page59", "dickson-theory-of-equations-1922/ex-page62", "dickson-theory-of-equations-1922/ex-page64", "dickson-theory-of-equations-1922/ex-page66", "dickson-theory-of-equations-1922/ex-page68", "dickson-theory-of-equations-1922/ex-page69", "dickson-theory-of-equations-1922/ex-page70" ] }, { "id": "dickson-theory-of-equations-1922/ch-vi", "number": "VI", "title": "Isolation of the Real Roots of a Real Equation", "name": "Dickson 1922, ch. VI: Isolation of the Real Roots of a Real Equation", "pages": [ "71", "86" ], "concepts": [ "concept/bend-point", "concept/coefficient", "concept/cubic-equation", "concept/degree", "concept/derivative", "concept/discriminant", "concept/graph-of-a-function", "concept/higher-order-derivative", "concept/highest-common-factor", "concept/imaginary-root", "concept/interval", "concept/multiple-root", "concept/polynomial", "concept/quartic-equation", "concept/real-root", "concept/reduced-quartic-equation", "concept/root-of-an-equation", "concept/sign", "concept/sturm-s-functions", "concept/variation-of-sign", "method/isolation-of-a-root", "theorem/budan-s-theorem", "theorem/descartes-rule-of-signs", "theorem/rolle-s-theorem", "theorem/sturm-s-theorem", "theorem/taylor-s-theorem" ], "excerpts": [ "dickson-theory-of-equations-1922/x-4f17717f9e", "dickson-theory-of-equations-1922/x-fb1bd97f16", "dickson-theory-of-equations-1922/x-f6648aee9d", "dickson-theory-of-equations-1922/x-12c4c34000", "dickson-theory-of-equations-1922/x-c9944bcbfb", "dickson-theory-of-equations-1922/x-58569773b3", "dickson-theory-of-equations-1922/x-364ec5bdb9", "dickson-theory-of-equations-1922/x-c0efe78486" ], "equations": [ "dickson-theory-of-equations-1922/eq-bff8156735", "dickson-theory-of-equations-1922/eq-a2fcb3e3ba", "dickson-theory-of-equations-1922/eq-b3b77d9f0f", "dickson-theory-of-equations-1922/eq-1eccab8397", "dickson-theory-of-equations-1922/eq-ec76e989a8", "dickson-theory-of-equations-1922/eq-02a3690bff", "dickson-theory-of-equations-1922/eq-3d4b8d70fb", "dickson-theory-of-equations-1922/eq-4ece3b817e", "dickson-theory-of-equations-1922/eq-dce102a460", "dickson-theory-of-equations-1922/eq-688575c9b0", "dickson-theory-of-equations-1922/eq-22c1a91c9f", "dickson-theory-of-equations-1922/eq-777063e486", "dickson-theory-of-equations-1922/eq-f0f118b2a3", "dickson-theory-of-equations-1922/eq-1752c87f49", "dickson-theory-of-equations-1922/eq-d66f92d186", "dickson-theory-of-equations-1922/eq-4e68fe907a", "dickson-theory-of-equations-1922/eq-1eb93a9c1c", "dickson-theory-of-equations-1922/eq-b8b56945d9", "dickson-theory-of-equations-1922/eq-bc6db44bd5", "dickson-theory-of-equations-1922/eq-34c02cbed8", "dickson-theory-of-equations-1922/eq-a16b8fc2d7", "dickson-theory-of-equations-1922/eq-7091be904d", "dickson-theory-of-equations-1922/eq-185a40dbe0", "dickson-theory-of-equations-1922/eq-03da7a5a48", "dickson-theory-of-equations-1922/eq-e3d047fc4d", "dickson-theory-of-equations-1922/eq-50e7b4244d", "dickson-theory-of-equations-1922/eq-39cd545676" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page74", "dickson-theory-of-equations-1922/ex-page78", "dickson-theory-of-equations-1922/ex-page79", "dickson-theory-of-equations-1922/ex-page81", "dickson-theory-of-equations-1922/ex-page83", "dickson-theory-of-equations-1922/ex-page85" ] }, { "id": "dickson-theory-of-equations-1922/ch-vii", "number": "VII", "title": "Solution of Numerical Equations", "name": "Dickson 1922, ch. VII: Solution of Numerical Equations", "pages": [ "86", "101" ], "concepts": [ "concept/bend-point", "concept/derivative", "concept/imaginary-root", "concept/inflection-point", "concept/polynomial", "concept/real-root", "concept/remainder", "concept/root-of-an-equation", "concept/subtangent", "concept/transformed-equation", "method/horner-s-method", "method/interpolating-in-logarithmic-tables", "method/isolation-of-a-root", "method/newton-s-method", "method/regula-falsi", "method/synthetic-division", "person/isaac-newton", "person/william-george-horner", "theorem/taylor-s-theorem", "unit/radian" ], "excerpts": [ "dickson-theory-of-equations-1922/x-91d521a554", "dickson-theory-of-equations-1922/x-ef66ec0ef1", "dickson-theory-of-equations-1922/x-5ded088744", "dickson-theory-of-equations-1922/x-92b26befd5", "dickson-theory-of-equations-1922/x-b113785214", "dickson-theory-of-equations-1922/x-09a9a335ac", "dickson-theory-of-equations-1922/x-b88978a295" ], "equations": [ "dickson-theory-of-equations-1922/eq-341c4382d1", "dickson-theory-of-equations-1922/eq-070613b1b8", "dickson-theory-of-equations-1922/eq-d49407fce0", "dickson-theory-of-equations-1922/eq-34a5c32978", "dickson-theory-of-equations-1922/eq-31c3fd2191", "dickson-theory-of-equations-1922/eq-64b87a678e", "dickson-theory-of-equations-1922/eq-78e020cb16", "dickson-theory-of-equations-1922/eq-4f85f5694d", "dickson-theory-of-equations-1922/eq-b0ab76ea7d", "dickson-theory-of-equations-1922/eq-2bc1e14efe", "dickson-theory-of-equations-1922/eq-c3a3422621", "dickson-theory-of-equations-1922/eq-87ade65e4d", "dickson-theory-of-equations-1922/eq-06f2dbb720", "dickson-theory-of-equations-1922/eq-9bc7fc81a4", "dickson-theory-of-equations-1922/eq-b09e3bab8b", "dickson-theory-of-equations-1922/eq-711f7fe774", "dickson-theory-of-equations-1922/eq-ec8c48a589", "dickson-theory-of-equations-1922/eq-f8a5d1b4f4", "dickson-theory-of-equations-1922/eq-9f4b8e5344", "dickson-theory-of-equations-1922/eq-a06b690dff", "dickson-theory-of-equations-1922/eq-e3f521bfae", "dickson-theory-of-equations-1922/eq-d2b9c97056", "dickson-theory-of-equations-1922/eq-5e64076267", "dickson-theory-of-equations-1922/eq-c437bf27d5", "dickson-theory-of-equations-1922/eq-feb66f0bea" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page89", "dickson-theory-of-equations-1922/ex-page94", "dickson-theory-of-equations-1922/ex-page96", "dickson-theory-of-equations-1922/ex-page98", "dickson-theory-of-equations-1922/ex-page99", "dickson-theory-of-equations-1922/ex-page100" ] }, { "id": "dickson-theory-of-equations-1922/ch-viii", "number": "VIII", "title": "Determinants; Systems of Linear Equations", "name": "Dickson 1922, ch. VIII: Determinants; Systems of Linear Equations", "pages": [ "114", "128" ], "concepts": [ "concept/augmented-matrix", "concept/circle", "concept/coefficient", "concept/complementary-minor", "concept/cube-root-of-unity", "concept/cubic-equation", "concept/determinant", "concept/diagonal-term-of-a-determinant", "concept/element-of-a-determinant", "concept/factor", "concept/homogeneous-linear-equations", "concept/known-term", "concept/linear-equation", "concept/matrix", "concept/minor", "concept/rank-of-a-determinant", "concept/simultaneous-equations", "concept/unknown", "method/laplace-s-development", "method/solving-simultaneous-equations-by-determinants", "person/gabriel-cramer", "theorem/consistency-of-a-linear-system", "theorem/cramer-s-rule", "theorem/product-of-determinants" ], "excerpts": [ "dickson-theory-of-equations-1922/x-a56f1131a1", "dickson-theory-of-equations-1922/x-68c56e1822", "dickson-theory-of-equations-1922/x-80050f4a42", "dickson-theory-of-equations-1922/x-c0f99fab31", "dickson-theory-of-equations-1922/x-c0489c724d", "dickson-theory-of-equations-1922/x-02f6d06a67", "dickson-theory-of-equations-1922/x-6360bed985", "dickson-theory-of-equations-1922/x-eb95b57f20", "dickson-theory-of-equations-1922/x-8a150dc650", "dickson-theory-of-equations-1922/x-ee75d546a5", "dickson-theory-of-equations-1922/x-f07466ded6" ], "equations": [ "dickson-theory-of-equations-1922/eq-b55c9de054", "dickson-theory-of-equations-1922/eq-050cc17f0e", "dickson-theory-of-equations-1922/eq-c14e78caa6", "dickson-theory-of-equations-1922/eq-91408a1efd", "dickson-theory-of-equations-1922/eq-c62045401c", "dickson-theory-of-equations-1922/eq-cb38a63ed9", "dickson-theory-of-equations-1922/eq-2a32685556", "dickson-theory-of-equations-1922/eq-1e4d086572", "dickson-theory-of-equations-1922/eq-ce8cfb1608", "dickson-theory-of-equations-1922/eq-e1eb73394c", "dickson-theory-of-equations-1922/eq-db2ca6317f", "dickson-theory-of-equations-1922/eq-ffb0d502b1", "dickson-theory-of-equations-1922/eq-62f1e40ac4", "dickson-theory-of-equations-1922/eq-a512c0f2f4", "dickson-theory-of-equations-1922/eq-f7988def99", "dickson-theory-of-equations-1922/eq-15b4864de1", "dickson-theory-of-equations-1922/eq-78c942b9d6", "dickson-theory-of-equations-1922/eq-a354ca37fe", "dickson-theory-of-equations-1922/eq-df8a4e28cd", "dickson-theory-of-equations-1922/eq-e9aa4680d4", "dickson-theory-of-equations-1922/eq-b971437ccd", "dickson-theory-of-equations-1922/eq-ce9c5ced09", "dickson-theory-of-equations-1922/eq-daab0d8ccf", "dickson-theory-of-equations-1922/eq-497c76589d", "dickson-theory-of-equations-1922/eq-dbf1e7bd6c", "dickson-theory-of-equations-1922/eq-5e61aafa7e", "dickson-theory-of-equations-1922/eq-f135ad4b00", "dickson-theory-of-equations-1922/eq-abfb6d9f72", "dickson-theory-of-equations-1922/eq-dbac901c70", "dickson-theory-of-equations-1922/eq-5ad187574c" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page115", "dickson-theory-of-equations-1922/ex-page119", "dickson-theory-of-equations-1922/ex-page102", "dickson-theory-of-equations-1922/ex-page104", "dickson-theory-of-equations-1922/ex-page106", "dickson-theory-of-equations-1922/ex-page108", "dickson-theory-of-equations-1922/ex-page112", "dickson-theory-of-equations-1922/ex-page113", "dickson-theory-of-equations-1922/ex-page120", "dickson-theory-of-equations-1922/ex-page121", "dickson-theory-of-equations-1922/ex-page124", "dickson-theory-of-equations-1922/ex-page125", "dickson-theory-of-equations-1922/ex-page126" ] }, { "id": "dickson-theory-of-equations-1922/ch-ix", "number": "IX", "title": "Symmetric Functions", "name": "Dickson 1922, ch. IX: Symmetric Functions", "pages": [ "128", "143" ], "concepts": [ "concept/elementary-symmetric-function", "concept/permutation", "concept/rational-function", "concept/root-of-an-equation", "concept/sigma-function", "concept/symmetric-function", "theorem/relations-between-roots-and-coefficients" ], "excerpts": [ "dickson-theory-of-equations-1922/x-c7df95e232", "dickson-theory-of-equations-1922/x-cbdf831b14", "dickson-theory-of-equations-1922/x-f32340740c", "dickson-theory-of-equations-1922/x-89f475ee0a" ], "equations": [ "dickson-theory-of-equations-1922/eq-e2f3df54b3", "dickson-theory-of-equations-1922/eq-081f76d03a", "dickson-theory-of-equations-1922/eq-6265949a3f", "dickson-theory-of-equations-1922/eq-e1b2f25eee", "dickson-theory-of-equations-1922/eq-b11e2ec77a", "dickson-theory-of-equations-1922/eq-fb80718189", "dickson-theory-of-equations-1922/eq-3ba551ef85", "dickson-theory-of-equations-1922/eq-f34f9cb43a", "dickson-theory-of-equations-1922/eq-d24e436492", "dickson-theory-of-equations-1922/eq-08c8145195", "dickson-theory-of-equations-1922/eq-9583c8ffe0", "dickson-theory-of-equations-1922/eq-27a706ae34", "dickson-theory-of-equations-1922/eq-c37d7f0687", "dickson-theory-of-equations-1922/eq-188bc48bb7", "dickson-theory-of-equations-1922/eq-e672c0143d", "dickson-theory-of-equations-1922/eq-48e43dbe19", "dickson-theory-of-equations-1922/eq-e95fa6bad7", "dickson-theory-of-equations-1922/eq-01e3548216", "dickson-theory-of-equations-1922/eq-b033c9c0af", "dickson-theory-of-equations-1922/eq-1206e991cb", "dickson-theory-of-equations-1922/eq-755ef18d99", "dickson-theory-of-equations-1922/eq-bcc9d28aef", "dickson-theory-of-equations-1922/eq-ca54a59572", "dickson-theory-of-equations-1922/eq-9aee83ffb8", "dickson-theory-of-equations-1922/eq-d0d7f21748", "dickson-theory-of-equations-1922/eq-8362dad116", "dickson-theory-of-equations-1922/eq-9192ea45fc", "dickson-theory-of-equations-1922/eq-ef9257bddf" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page129", "dickson-theory-of-equations-1922/ex-page133", "dickson-theory-of-equations-1922/ex-page136", "dickson-theory-of-equations-1922/ex-page140", "dickson-theory-of-equations-1922/ex-page141", "dickson-theory-of-equations-1922/ex-page142" ] }, { "id": "dickson-theory-of-equations-1922/ch-x", "number": "X", "title": "Elimination, Resultants And Discriminants", "name": "Dickson 1922, ch. X: Elimination, Resultants And Discriminants", "pages": [ "143", "155" ], "concepts": [ "concept/coefficient", "concept/common-root", "concept/conic-section", "concept/cubic-equation", "concept/degree", "concept/determinant", "concept/discriminant", "concept/elementary-symmetric-function", "concept/extraneous-factor", "concept/homogeneous-linear-equations", "concept/multiple-root", "concept/polynomial", "concept/resultant", "concept/root-of-an-equation", "method/b-zout-s-method-of-elimination", "method/elimination", "method/euler-s-method-of-elimination", "method/laplace-s-development", "method/sylvester-s-dialytic-method", "person/james-joseph-sylvester", "person/leonhard-euler", "person/tienne-b-zout" ], "excerpts": [ "dickson-theory-of-equations-1922/x-27eb0767d3", "dickson-theory-of-equations-1922/x-697ac048f9", "dickson-theory-of-equations-1922/x-f7317aa938", "dickson-theory-of-equations-1922/x-7a46dec5b9", "dickson-theory-of-equations-1922/x-e155ab5f83", "dickson-theory-of-equations-1922/x-68501f81c6" ], "equations": [ "dickson-theory-of-equations-1922/eq-99dd9da92e", "dickson-theory-of-equations-1922/eq-a3f632d227", "dickson-theory-of-equations-1922/eq-1c592bef2d", "dickson-theory-of-equations-1922/eq-8dc4c6294c", "dickson-theory-of-equations-1922/eq-576fe1c9bf", "dickson-theory-of-equations-1922/eq-f99c612e6b", "dickson-theory-of-equations-1922/eq-34898b9e96", "dickson-theory-of-equations-1922/eq-26bf9eb023", "dickson-theory-of-equations-1922/eq-b511a4cb80", "dickson-theory-of-equations-1922/eq-4c32a6fa99", "dickson-theory-of-equations-1922/eq-8706881e8c", "dickson-theory-of-equations-1922/eq-5bef72c8bf", "dickson-theory-of-equations-1922/eq-2370e9c8e6", "dickson-theory-of-equations-1922/eq-a7cdbc3f9b", "dickson-theory-of-equations-1922/eq-11dd105fd5", "dickson-theory-of-equations-1922/eq-23d142c83a", "dickson-theory-of-equations-1922/eq-12e3237baa" ], "exercise_sets": [ "dickson-theory-of-equations-1922/ex-page144", "dickson-theory-of-equations-1922/ex-page147", "dickson-theory-of-equations-1922/ex-page150", "dickson-theory-of-equations-1922/ex-page152", "dickson-theory-of-equations-1922/ex-page153b", "dickson-theory-of-equations-1922/ex-page153" ] }, { "id": "dickson-theory-of-equations-1922/ch-appendix", "number": "Appendix", "title": "Appendix", "name": "Dickson 1922, Appendix", "pages": [ "155", "158" ], "concepts": [ "concept/abscissa", "concept/absolute-value", "concept/circle", "concept/complex-number", "concept/continuous-function", "concept/curve", "concept/degree", "concept/function-of-two-variables", "concept/graph-of-a-function", "concept/minimum", "concept/polynomial", "concept/proof", "concept/right-circular-cylinder", "concept/root", "concept/trigonometric-form-of-a-complex-number", "person/augustin-louis-cauchy", "person/carl-friedrich-gauss", "quantity/modulus-of-a-complex-number", "theorem/binomial-theorem", "theorem/fundamental-theorem-of-algebra", "theorem/taylor-s-theorem" ], "excerpts": [ "dickson-theory-of-equations-1922/x-96c7d33a88", "dickson-theory-of-equations-1922/x-e7c12b3a70", "dickson-theory-of-equations-1922/x-e82f9974dc", "dickson-theory-of-equations-1922/x-7d58af1318", "dickson-theory-of-equations-1922/x-da3afa5ffb" ], "equations": [ "dickson-theory-of-equations-1922/eq-2e8f3c53ee", "dickson-theory-of-equations-1922/eq-bcdc18a1be", "dickson-theory-of-equations-1922/eq-cde9467aee", "dickson-theory-of-equations-1922/eq-4094994b57", "dickson-theory-of-equations-1922/eq-57f0b061cd", "dickson-theory-of-equations-1922/eq-763906457f", "dickson-theory-of-equations-1922/eq-63021b5876", "dickson-theory-of-equations-1922/eq-121cb70165", "dickson-theory-of-equations-1922/eq-55fdbd4dce", "dickson-theory-of-equations-1922/eq-aae1ab37e0", "dickson-theory-of-equations-1922/eq-ab0d37c77b", "dickson-theory-of-equations-1922/eq-d198694967", "dickson-theory-of-equations-1922/eq-7b5eabbee8", "dickson-theory-of-equations-1922/eq-ea00ba0510", "dickson-theory-of-equations-1922/eq-3cc9f23397", "dickson-theory-of-equations-1922/eq-b80fa895a6", "dickson-theory-of-equations-1922/eq-448e97d4c4", "dickson-theory-of-equations-1922/eq-47be3a5cd5", "dickson-theory-of-equations-1922/eq-cbdb6ee0b7", "dickson-theory-of-equations-1922/eq-b365c1e9da", "dickson-theory-of-equations-1922/eq-802dd70860", "dickson-theory-of-equations-1922/eq-c13a38941d", "dickson-theory-of-equations-1922/eq-7ae83c12ce", "dickson-theory-of-equations-1922/eq-a0687933dd", "dickson-theory-of-equations-1922/eq-188881ec7f", "dickson-theory-of-equations-1922/eq-538016afb8", "dickson-theory-of-equations-1922/eq-ca1ad27dcb" ], "exercise_sets": [] } ], "excerpts": [ { "id": "dickson-theory-of-equations-1922/x-a05f5e1381", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "is called a \\emph{quadratic equation} or equation of the second degree.", "markdown": "is called a *quadratic equation* or equation of the second degree.", "why": "Gives the learner the defining sentence for the quadratic equation before any solving begins.", "use": [ "lesson" ], "concepts": [ "concept/quadratic-equation" ] }, { "id": "dickson-theory-of-equations-1922/x-8ad9ce7628", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "12", "location": "Elementary Theorems on the Roots of an Equation", "latex": "If a polynomial $f(x)$ be divided by $x - c$ until a remainder independent of~$x$ is obtained, this remainder is equal to~$f(c)$, which is the value of~$f(x)$ when $x = c$.", "markdown": "If a polynomial $f(x)$ be divided by $x - c$ until a remainder independent of $x$ is obtained, this remainder is equal to $f(c)$, which is the value of $f(x)$ when $x = c$.", "why": "States the remainder theorem so a learner can check a value by division instead of substitution.", "use": [ "lesson", "website" ], "concepts": [ "theorem/remainder-theorem" ] }, { "id": "dickson-theory-of-equations-1922/x-e00a7f71e7", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "12", "location": "Elementary Theorems on the Roots of an Equation", "latex": "If $f(c)$ is zero, the polynomial $f(x)$ has the factor $x - c$. In other words, if $c$ is a root of $f(x) = 0$, $x - c$ is a factor of~$f(x)$.", "markdown": "If $f(c)$ is zero, the polynomial $f(x)$ has the factor $x - c$. In other words, if $c$ is a root of $f(x) = 0$, $x - c$ is a factor of $f(x)$.", "why": "Links roots and factors so a learner can find a factor once a root is known.", "use": [ "lesson" ], "concepts": [ "concept/factor", "concept/root-of-an-equation", "theorem/factor-theorem" ] }, { "id": "dickson-theory-of-equations-1922/x-25f2cde19e", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "14", "location": "Elementary Theorems on the Roots of an Equation", "latex": "First we bring down the first coefficient~$1$. Then we multiply it by the given value~$2$ and enter the product~$2$ directly under the second coefficient~$3$, add and write the sum~$5$ below.", "markdown": "First we bring down the first coefficient $1$. Then we multiply it by the given value $2$ and enter the product $2$ directly under the second coefficient $3$, add and write the sum $5$ below.", "why": "A step-by-step worked start of synthetic division that a learner can follow with pencil.", "use": [ "lesson" ], "concepts": [ "method/synthetic-division" ] }, { "id": "dickson-theory-of-equations-1922/x-b35ac51792", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "12", "location": "Elementary Theorems on the Roots of an Equation", "latex": "We thus obtain the useful result that $ax^2 + bx + c$ \\emph{is a perfect square (of a linear function of $x$) if and only if $b^2 = 4ac$ (\\emph{i.e.}, if its discriminant is zero)}.", "markdown": "We thus obtain the useful result that $ax^2 + bx + c$ *is a perfect square (of a linear function of $x$) if and only if $b^2 = 4ac$ (*i.e.*, if its discriminant is zero)*.", "why": "Shows how the discriminant tests whether a quadratic is a perfect square, a useful check learners often need.", "use": [ "lesson" ], "concepts": [ "concept/discriminant", "concept/perfect-square" ] }, { "id": "dickson-theory-of-equations-1922/x-4fe43fe1cf", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "16", "location": "Elementary Theorems on the Roots of an Equation", "latex": "An equation of degree~$n$ cannot have more than $n$~roots, a root of multiplicity~$m$ being counted as $m$~roots.", "markdown": "An equation of degree $n$ cannot have more than $n$ roots, a root of multiplicity $m$ being counted as $m$ roots.", "why": "Explains why a repeated root must be counted with its multiplicity when counting roots.", "use": [ "lesson" ], "concepts": [ "concept/multiple-root", "theorem/equation-of-degree-n-has-at-most-n-roots" ] }, { "id": "dickson-theory-of-equations-1922/x-9f8bfb4329", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "21", "location": "Elementary Theorems on the Roots of an Equation", "latex": "For example, in $x^5 + 4x^4 - 7x^2 - 40x + 1 = 0$, $G = 40$ and $k = 3$ since we must supply the coefficient zero to the missing power~$x^3$.", "markdown": "For example, in $x^5 + 4x^4 - 7x^2 - 40x + 1 = 0$, $G = 40$ and $k = 3$ since we must supply the coefficient zero to the missing power $x^3$.", "why": "A concrete example showing how a missing power is counted as a zero coefficient when bounding roots.", "use": [ "lesson" ], "concepts": [ "concept/upper-limit-to-the-roots", "theorem/upper-limit-theorem-by-greatest-negative-coefficient" ] }, { "id": "dickson-theory-of-equations-1922/x-e28b954deb", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "The product of the squares of the differences of the roots of any equation in which the coefficient of the highest power of the unknown is unity shall be called the \\emph{discriminant} of the equation.", "markdown": "The product of the squares of the differences of the roots of any equation in which the coefficient of the highest power of the unknown is unity shall be called the *discriminant* of the equation.", "why": "Gives the learner the exact definition of the discriminant before any formula is used.", "use": [ "lesson", "website" ], "concepts": [ "concept/discriminant" ] }, { "id": "dickson-theory-of-equations-1922/x-9c839e64b5", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "A cubic equation with real coefficients has three distinct real roots if its discriminant~$\\Delta$ is positive, a single real root and two conjugate imaginary roots if $\\Delta$ is negative, and at least two equal real roots if $\\Delta$ is zero.", "markdown": "A cubic equation with real coefficients has three distinct real roots if its discriminant $\\Delta$ is positive, a single real root and two conjugate imaginary roots if $\\Delta$ is negative, and at least two equal real roots if $\\Delta$ is zero.", "why": "Shows how one number, the sign of the discriminant, tells a student how many real roots a cubic has.", "use": [ "lesson" ], "concepts": [ "concept/discriminant", "theorem/number-of-real-roots-of-a-cubic" ] }, { "id": "dickson-theory-of-equations-1922/x-f8e49eaff1", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "This is called the irreducible case since it may be shown that a cube root of a general complex number cannot be expressed in the form $a + bi$, where $a$ and $b$ involve only real radicals.", "markdown": "This is called the irreducible case since it may be shown that a cube root of a general complex number cannot be expressed in the form $a + bi$, where $a$ and $b$ involve only real radicals.", "why": "Explains why three real roots can still require imaginary quantities in the algebraic formula, which is the source of the name.", "use": [ "lesson" ], "concepts": [ "concept/irreducible-case" ] }, { "id": "dickson-theory-of-equations-1922/x-b37e967bc0", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "The expression $A + B$ for a root was first published by Cardan in his \\textit{Ars Magna} of~1545, although he had obtained it from Tartaglia under promise of secrecy.", "markdown": "The expression $A + B$ for a root was first published by Cardan in his *Ars Magna* of 1545, although he had obtained it from Tartaglia under promise of secrecy.", "why": "Credits the formula to its publisher and notes that it came from Tartaglia, a useful history of how the result was found.", "use": [ "history" ], "concepts": [ "person/girolamo-cardano", "theorem/cardan-s-formulas" ] }, { "id": "dickson-theory-of-equations-1922/x-234ee0c0f6", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "The pairs of values of~$z$ whose product is $5$ are $1$ and~$5$, $\\omega$ and $5\\omega^2$, $\\omega^2$ and $5\\omega$.", "markdown": "The pairs of values of $z$ whose product is $5$ are $1$ and $5$, $\\omega$ and $5\\omega^2$, $\\omega^2$ and $5\\omega$.", "why": "Shows the pairing step of Cardan's method on a concrete cubic, where each pair's product is fixed as -p/3.", "use": [ "lesson" ], "concepts": [ "concept/reduced-cubic-equation", "theorem/cardan-s-formulas" ] }, { "id": "dickson-theory-of-equations-1922/x-e5ac8e5fc0", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "This expression (and not $P$ itself) is called the discriminant of~\\Eq{13}.", "markdown": "This expression (and not $P$ itself) is called the discriminant of $(13)$.", "why": "Warns that for a cubic whose leading coefficient is not 1, the discriminant is the scaled expression and not the product of squared differences.", "use": [ "lesson" ], "concepts": [ "concept/discriminant" ] }, { "id": "dickson-theory-of-equations-1922/x-ce027671f0", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "50", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "Choose any root~$y$ of this \\emph{resolvent cubic equation~\\Eq{17}}. Then the right member of~\\Eq{16} is the square of a linear function, say $mx+n$.", "markdown": "Choose any root $y$ of this *resolvent cubic equation $(17)$*. Then the right member of $(16)$ is the square of a linear function, say $mx+n$.", "why": "Shows the key step of Ferrari's method, where a root of the resolvent cubic makes one side a perfect square.", "use": [ "lesson" ], "concepts": [ "concept/resolvent-cubic", "method/ferrari-s-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/x-d711aab631", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "53", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "The value $k^2 = 1$ gives the factors $z^2 + 2z - 1$, $z^2 - 2z + 2$.", "markdown": "The value $k^2 = 1$ gives the factors $z^2 + 2z - 1$, $z^2 - 2z + 2$.", "why": "A worked numerical instance of Descartes' factoring, which a learner can check by multiplying the factors back out.", "use": [ "lesson" ], "concepts": [ "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/x-41a01ff24a", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "56", "location": "The Graph of an Equation", "latex": "A point (like $M$ or~$M'$ in Fig.~14) is called a \\emph{bend point} of the graph of \\index{Bend point}% $y=f(x)$ if the tangent to the graph at that point is horizontal and if all of the adjacent points of the graph lie below the tangent or all above the tangent.", "markdown": "A point (like $M$ or $M'$ in Fig. 14) is called a *bend point* of the graph of % $y=f(x)$ if the tangent to the graph at that point is horizontal and if all of the adjacent points of the graph lie below the tangent or all above the tangent.", "why": "It defines the bend point precisely, so a learner can tell it apart from a point that only looks like a turning point.", "use": [ "lesson" ], "concepts": [ "concept/bend-point" ] }, { "id": "dickson-theory-of-equations-1922/x-a93a96ffb4", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "57", "location": "The Graph of an Equation", "latex": "We call $3x^2 + 8x$ the \\emph{derivative} of $x^3 + 4x^2 - 11$.", "markdown": "We call $3x^2 + 8x$ the *derivative* of $x^3 + 4x^2 - 11$.", "why": "It names the derivative as the limit of the secant slope, with a concrete polynomial a learner can check by hand.", "use": [ "lesson" ], "concepts": [ "concept/derivative" ] }, { "id": "dickson-theory-of-equations-1922/x-aaaf2f808e", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "The Graph of an Equation", "latex": "Thus the derivative of $a_0 x^n$ is $na_0 x^{n-1}$, and hence is obtained by multiplying the given term by its exponent~$n$ and then diminishing its exponent by unity.", "markdown": "Thus the derivative of $a_0 x^n$ is $na_0 x^{n-1}$, and hence is obtained by multiplying the given term by its exponent $n$ and then diminishing its exponent by unity.", "why": "It states the power rule as a procedure a learner can apply term by term.", "use": [ "lesson" ], "concepts": [ "method/differentiation" ] }, { "id": "dickson-theory-of-equations-1922/x-7e0fc26cea", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "56", "location": "The Graph of an Equation", "latex": "The true curve between two points below the $x$-axis may not cross the $x$-axis, or may have a peak and actually cross the $x$-axis twice, or may be an \\Shape{M}-shaped curve crossing it four times, etc.", "markdown": "The true curve between two points below the $x$-axis may not cross the $x$-axis, or may have a peak and actually cross the $x$-axis twice, or may be an M-shaped curve crossing it four times, etc.", "why": "It warns that a curve sketched through a few computed points can hide roots, which is why bend points are used.", "use": [ "lesson", "website" ], "concepts": [] }, { "id": "dickson-theory-of-equations-1922/x-6e76e36cf4", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "The Graph of an Equation", "latex": "This formula~\\Eq{8} is known as \\emph{Taylor's theorem} for the present case of \\index{Taylor's theorem}% a polynomial~$f(x)$ of degree~$n$.", "markdown": "This formula $(8)$ is known as *Taylor’s theorem* for the present case of % a polynomial $f(x)$ of degree $n$.", "why": "It names the expansion that gives every derivative of a polynomial at once, and it is the source of the derivative coefficients.", "use": [ "history" ], "concepts": [] }, { "id": "dickson-theory-of-equations-1922/x-6830ce5423", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "55", "location": "The Graph of an Equation", "latex": "To find geometrically the real roots of a real equation $f(x)=0$, we construct a graph of $y=f(x)$ and measure the distances from the origin~$O$ to the intersections of the graph and the $x$-axis, whose equation is $y=0$.", "markdown": "To find geometrically the real roots of a real equation $f(x)=0$, we construct a graph of $y=f(x)$ and measure the distances from the origin $O$ to the intersections of the graph and the $x$-axis, whose equation is $y=0$.", "why": "It states the whole idea of solving an equation by graphing in one sentence a learner can carry away.", "use": [ "lesson", "website" ], "concepts": [ "concept/graph-of-a-function", "concept/origin", "concept/real-root", "method/graphical-solution-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/x-fb53dcb8e9", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "56", "location": "The Graph of an Equation", "latex": "The purpose of the example was, however, not to point out this obvious fact, but rather to emphasize the chance of serious error in sketching a curve through a number of points, however numerous.", "markdown": "The purpose of the example was, however, not to point out this obvious fact, but rather to emphasize the chance of serious error in sketching a curve through a number of points, however numerous.", "why": "It warns learners that plotting a few points can mislead them about the true shape of a graph.", "use": [ "lesson" ], "concepts": [ "concept/graph-of-a-function" ] }, { "id": "dickson-theory-of-equations-1922/x-d3c80f6dcb", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "60", "location": "The Graph of an Equation", "latex": "In this sense the tangent at~$O$ is said to meet the curve in three coincident points, their abscissas being the three coinciding roots of $x^3 = 0$.", "markdown": "In this sense the tangent at $O$ is said to meet the curve in three coincident points, their abscissas being the three coinciding roots of $x^3 = 0$.", "why": "It gives a vivid picture of a triple root as three points of intersection merging into one.", "use": [ "lesson", "website" ], "concepts": [ "concept/abscissa", "concept/multiple-root", "concept/tangent" ] }, { "id": "dickson-theory-of-equations-1922/x-b64fc54d3f", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "65", "location": "The Graph of an Equation", "latex": "Hence \\emph{$x^3 - 3lx + q = 0$ has three distinct real roots if and only if $q^2 < 4l^3$, a single real root if and only if $q^2 > 4l^3$, a double root \\(necessarily real\\) if and only if $q^2 = 4l^3$ and $l\\neq 0$, and a triple root if $q^2 = 4l^3 = 0$}.", "markdown": "Hence *$x^3 - 3lx + q = 0$ has three distinct real roots if and only if $q^2 < 4l^3$, a single real root if and only if $q^2 > 4l^3$, a double root necessarily real if and only if $q^2 = 4l^3$ and $l\\neq 0$, and a triple root if $q^2 = 4l^3 = 0$*.", "why": "It states in one result how the sign of q^2 - 4l^3 decides the number of real roots of a cubic.", "use": [ "lesson" ], "concepts": [ "concept/cubic-equation", "concept/multiple-root", "concept/real-root" ] }, { "id": "dickson-theory-of-equations-1922/x-adb0691fc0", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "69", "location": "The Graph of an Equation", "latex": "Between two consecutive real roots $a$ and~$b$ of $f(x)=0$, there is an odd number of real roots of $f'(x) = 0$, a root of multiplicity~$m$ being counted as $m$~roots.", "markdown": "Between two consecutive real roots $a$ and $b$ of $f(x)=0$, there is an odd number of real roots of $f'(x) = 0$, a root of multiplicity $m$ being counted as $m$ roots.", "why": "It gives learners the link between the roots of a polynomial and those of its derivative, which is the key to counting roots.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "concept/multiple-root", "concept/real-root", "theorem/rolle-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/x-ea7d4d2020", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "61", "location": "The Graph of an Equation", "latex": "For example, $x^4 + 2x^3 =0$ has the triple root $x = 0$ since $0$ is a root, and since the first and second derivatives $4x^3 +6x^2$ and $12x^2 +12x$ are zero for $x = 0$, while the third derivative $24x + 12$ is not zero for $x = 0$.", "markdown": "For example, $x^4 + 2x^3 =0$ has the triple root $x = 0$ since $0$ is a root, and since the first and second derivatives $4x^3 +6x^2$ and $12x^2 +12x$ are zero for $x = 0$, while the third derivative $24x + 12$ is not zero for $x = 0$.", "why": "It works a complete example showing how derivatives detect the multiplicity of a root.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "concept/higher-order-derivative", "concept/multiple-root" ] }, { "id": "dickson-theory-of-equations-1922/x-3bf441d44c", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "58", "location": "The Graph of an Equation", "latex": "The use of the bend points insures greater accuracy to the graph than the use of dozens of points whose abscissas are taken at random.", "markdown": "The use of the bend points insures greater accuracy to the graph than the use of dozens of points whose abscissas are taken at random.", "why": "It makes the case for locating bend points over plotting many arbitrary points, in the author's plain old-fashioned voice.", "use": [ "website", "lesson" ], "concepts": [ "concept/abscissa", "concept/bend-point", "concept/graph-of-a-function" ] }, { "id": "dickson-theory-of-equations-1922/x-4f17717f9e", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "71", "location": "Isolation of the Real Roots of a Real Equation", "latex": "But in the contrary case, narrower limits are necessary, such\nas $4$ and~$4.3$, with the further fact that only one root is between these new\nlimits. Then that root is said to be \\emph{isolated}.", "markdown": "But in the contrary case, narrower limits are necessary, such as $4$ and $4.3$, with the further fact that only one root is between these new limits. Then that root is said to be *isolated*.", "why": "Defines isolation of a root by a concrete pair of limits, so a learner sees what 'isolated' means before any method is applied.", "use": [ "lesson", "website" ], "concepts": [ "concept/real-root", "method/isolation-of-a-root" ] }, { "id": "dickson-theory-of-equations-1922/x-fb1bd97f16", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "71", "location": "Isolation of the Real Roots of a Real Equation", "latex": "Thus, in $x^5 - 2x^3 - 4x^2 + 3 = 0$, the first two terms present a variation of sign, and\nlikewise the last two terms. The number of variations of sign of the equation is two.", "markdown": "Thus, in $x^5 - 2x^3 - 4x^2 + 3 = 0$, the first two terms present a variation of sign, and likewise the last two terms. The number of variations of sign of the equation is two.", "why": "Shows how to count variations of sign on a concrete equation, the step on which Descartes' rule depends.", "use": [ "lesson" ], "concepts": [ "concept/variation-of-sign", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/x-f6648aee9d", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "72", "location": "Isolation of the Real Roots of a Real Equation", "latex": "The number of positive real roots of an equation\nwith real coefficients is either equal to the number of its variations of sign\nor is less than that number by a positive even integer. A root of multiplicity~$m$\nis here counted as $m$~roots.", "markdown": "The number of positive real roots of an equation with real coefficients is either equal to the number of its variations of sign or is less than that number by a positive even integer. A root of multiplicity $m$ is here counted as $m$ roots.", "why": "States Descartes' rule exactly, including the convention that a multiple root is counted by its multiplicity.", "use": [ "lesson" ], "concepts": [ "concept/multiple-root", "concept/variation-of-sign", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/x-12c4c34000", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "72", "location": "Isolation of the Real Roots of a Real Equation", "latex": "For example, $x^6 - 3x^2 + x + 1 = 0$ has either two or no positive roots, the exact number\nnot being found. But $3x^3 - x - 1 = 0$ has exactly one positive root, which is a simple\nroot.", "markdown": "For example, $x^6 - 3x^2 + x + 1 = 0$ has either two or no positive roots, the exact number not being found. But $3x^3 - x - 1 = 0$ has exactly one positive root, which is a simple root.", "why": "Contrasts an equation where the rule leaves a choice with one where it gives an exact count, teaching what the rule can and cannot decide.", "use": [ "lesson" ], "concepts": [ "concept/multiple-root", "concept/real-root", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/x-c9944bcbfb", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "76", "location": "Isolation of the Real Roots of a Real Equation", "latex": "Experience shows that most students make some\nerror in finding $f_2, f_3, \\dotsc$, so that checking is essential.", "markdown": "Experience shows that most students make some error in finding $f_2, f_3, \\dotsc$, so that checking is essential.", "why": "Tells the learner to check each Sturm function by substitution, a habit that catches the most common slip.", "use": [ "lesson" ], "concepts": [ "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/x-58569773b3", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "71", "location": "Isolation of the Real Roots of a Real Equation", "latex": "Unfortunately it rarely tells us the exact number of real roots.", "markdown": "Unfortunately it rarely tells us the exact number of real roots.", "why": "An honest limit on Descartes' rule that prevents learners from treating it as a full count.", "use": [ "website" ], "concepts": [ "concept/real-root", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/x-364ec5bdb9", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "78", "location": "Isolation of the Real Roots of a Real Equation", "latex": "A violation of this Corollary usually indicates an error in the computation\nof Sturm's functions~\\Eq{2}.", "markdown": "A violation of this Corollary usually indicates an error in the computation of Sturm’s functions $(2)$.", "why": "Gives a self-check: if the variation count rises from left to right, the Sturm functions were computed wrongly.", "use": [ "lesson" ], "concepts": [ "concept/sturm-s-functions", "theorem/sturm-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/x-c0efe78486", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "72", "location": "Isolation of the Real Roots of a Real Equation", "latex": "Descartes' rule will be derived in §73 as a corollary to Budan's theorem.", "markdown": "Descartes’ rule will be derived in §73 as a corollary to Budan’s theorem.", "why": "Shows the historical logical order in which the book derives Descartes' rule from the more general Budan theorem.", "use": [ "history" ], "concepts": [ "theorem/budan-s-theorem", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/x-91d521a554", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "86", "location": "Solution of Numerical Equations", "latex": "Hence $-1$ is the remainder obtained when the given polynomial $x^3 - 2x - 5$ is divided by $x-2$.", "markdown": "Hence $-1$ is the remainder obtained when the given polynomial $x^3 - 2x - 5$ is divided by $x-2$.", "why": "It shows the learner why dividing by x-2 yields the coefficients of the shifted equation, the core idea of Horner's method.", "use": [ "lesson" ], "concepts": [ "concept/polynomial", "concept/remainder", "method/horner-s-method" ] }, { "id": "dickson-theory-of-equations-1922/x-ef66ec0ef1", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "91", "location": "Solution of Numerical Equations", "latex": "Newton used the close approximation $0.1$ to~$p$, in spite of the fact that this value exceeds the root~$p$ and hence led to a negative correction at the next step.", "markdown": "Newton used the close approximation $0.1$ to $p$, in spite of the fact that this value exceeds the root $p$ and hence led to a negative correction at the next step.", "why": "It makes the sign of the correction, and why an approximation above the root is still usable, visible to a learner.", "use": [ "lesson" ], "concepts": [ "concept/root-of-an-equation", "method/newton-s-method" ] }, { "id": "dickson-theory-of-equations-1922/x-5ded088744", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "91", "location": "Solution of Numerical Equations", "latex": "Given an approximate value~$a$ of a real root, we can usually find a closer approximation $a+h$ to the root by neglecting the powers $h^2$, $h^3, \\dotsc$ of the small number~$h$ in Taylor's formula~(§56)", "markdown": "Given an approximate value $a$ of a real root, we can usually find a closer approximation $a+h$ to the root by neglecting the powers $h^2$, $h^3, \\dotsc$ of the small number $h$ in Taylor’s formula (§56)", "why": "It states the idea behind Newton's method in one sentence: linearise with Taylor's formula and solve for the small correction.", "use": [ "lesson" ], "concepts": [ "method/newton-s-method", "theorem/taylor-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/x-92b26befd5", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "86", "location": "Solution of Numerical Equations", "latex": "W.~G. Horner, London Philosophical Transactions, 1819. Earlier (1804) by P.~Ruffini.", "markdown": "W. G. Horner, London Philosophical Transactions, 1819. Earlier (1804) by P. Ruffini.", "why": "It gives the historical credit for the method and notes an earlier publication, which is useful for a history reading.", "use": [ "history" ], "concepts": [ "method/horner-s-method", "person/william-george-horner" ] }, { "id": "dickson-theory-of-equations-1922/x-b113785214", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "92", "location": "Solution of Numerical Equations", "latex": "Failure is certain if we use a point~$P_2$ such that a single bend point lies between it and~$S$.", "markdown": "Failure is certain if we use a point $P_2$ such that a single bend point lies between it and $S$.", "why": "It warns the learner that a starting point on the far side of a bend can make Newton's method fail, so the starting point must be chosen with care.", "use": [ "lesson" ], "concepts": [ "concept/bend-point", "method/newton-s-method" ] }, { "id": "dickson-theory-of-equations-1922/x-09a9a335ac", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "93", "location": "Solution of Numerical Equations", "latex": "The advantage of having $c$ at each step is that we know a close limit of the error made in the approximation to the root.", "markdown": "The advantage of having $c$ at each step is that we know a close limit of the error made in the approximation to the root.", "why": "It explains why bracketing methods are preferred when a guaranteed error bound matters.", "use": [ "lesson", "website" ], "concepts": [ "method/regula-falsi" ] }, { "id": "dickson-theory-of-equations-1922/x-b88978a295", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Solution of Numerical Equations", "latex": "To find the imaginary roots $x+yi$ of an equation $f(z)=0$ with real coefficients, expand $f(x+yi)$ by Taylor's theorem;", "markdown": "To find the imaginary roots $x+yi$ of an equation $f(z)=0$ with real coefficients, expand $f(x+yi)$ by Taylor’s theorem;", "why": "It shows how Taylor's theorem turns the search for complex roots into two real equations a learner can solve.", "use": [ "lesson" ], "concepts": [ "concept/imaginary-root", "theorem/taylor-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/x-a56f1131a1", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Determinants; Systems of Linear Equations", "latex": "If $D \\neq 0$, the unique values of $x_1, \\dotsc, x_n$ determined by division from~\\Eq{13} actually satisfy equations~\\Eq{12}.", "markdown": "If $D \\neq 0$, the unique values of $x_1, \\dotsc, x_n$ determined by division from $(13)$ actually satisfy equations $(12)$.", "why": "It tells the learner that the values found by dividing by D really do solve the original equations, not just the derived ones.", "use": [ "lesson" ], "concepts": [ "theorem/cramer-s-rule" ] }, { "id": "dickson-theory-of-equations-1922/x-68c56e1822", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Determinants; Systems of Linear Equations", "latex": "If $D$ denotes the determinant of the coefficients of the $n$~unknowns in a system of $n$~linear equations, the product of $D$ by any one of the unknowns is equal to the determinant obtained from $D$ by substituting the known terms in place of the coefficients of that unknown. If $D \\neq 0$, we obtain the unique values of the unknowns by division by~$D$.", "markdown": "If $D$ denotes the determinant of the coefficients of the $n$ unknowns in a system of $n$ linear equations, the product of $D$ by any one of the unknowns is equal to the determinant obtained from $D$ by substituting the known terms in place of the coefficients of that unknown. If $D \\neq 0$, we obtain the unique values of the unknowns by division by $D$.", "why": "It states Cramer's rule in one place, showing a learner how the determinant of the coefficients yields each unknown.", "use": [ "lesson", "website" ], "concepts": [ "concept/determinant", "concept/unknown", "theorem/cramer-s-rule" ] }, { "id": "dickson-theory-of-equations-1922/x-80050f4a42", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "101", "location": "Determinants; Systems of Linear Equations", "latex": "Multiply the members of the first equation by~$b_2$ and those of the second equation by~$-b_1$, and add the resulting equations.", "markdown": "Multiply the members of the first equation by $b_2$ and those of the second equation by $-b_1$, and add the resulting equations.", "why": "It shows the elimination step that produces the coefficient determinant, which a learner can follow with pencil and paper.", "use": [ "lesson" ], "concepts": [ "concept/simultaneous-equations", "method/solving-simultaneous-equations-by-determinants" ] }, { "id": "dickson-theory-of-equations-1922/x-c0f99fab31", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "101", "location": "Determinants; Systems of Linear Equations", "latex": "\\emph{if $D$ is the determinant of the coefficients of the unknowns, the product of~$D$ by any one of the unknowns is equal to the determinant obtained from~$D$ by substituting the known terms in place of the coefficients of that unknown}.", "markdown": "*if $D$ is the determinant of the coefficients of the unknowns, the product of $D$ by any one of the unknowns is equal to the determinant obtained from $D$ by substituting the known terms in place of the coefficients of that unknown*.", "why": "It states the rule in words, so the learner knows what to substitute for each unknown.", "use": [ "lesson" ], "concepts": [ "concept/determinant", "concept/known-term", "method/solving-simultaneous-equations-by-determinants" ] }, { "id": "dickson-theory-of-equations-1922/x-c0489c724d", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "103", "location": "Determinants; Systems of Linear Equations", "latex": "The nine numbers $a_1, \\dotsc, c_3$ are called the \\emph{elements} of the determinant.", "markdown": "The nine numbers $a_1, \\dotsc, c_3$ are called the *elements* of the determinant.", "why": "It fixes the vocabulary of elements, rows and columns before the reader meets the third-order symbol.", "use": [ "lesson" ], "concepts": [ "concept/determinant", "concept/element-of-a-determinant" ] }, { "id": "dickson-theory-of-equations-1922/x-02f6d06a67", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "102", "location": "Determinants; Systems of Linear Equations", "latex": "If $D \\ne 0$, relations~\\Eq{3} uniquely determine values of $x$ and~$y$:", "markdown": "If $D \\ne 0$, relations $(3)$ uniquely determine values of $x$ and $y$:", "why": "It states the condition under which the determinant method gives a unique answer, which a learner needs before applying it.", "use": [ "lesson" ], "concepts": [ "concept/determinant", "method/solving-simultaneous-equations-by-determinants" ] }, { "id": "dickson-theory-of-equations-1922/x-6360bed985", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Determinants; Systems of Linear Equations", "latex": "A system of $m$~linear equations in $n$~unknowns is consistent if and only if the rank of the matrix of the coefficients of the unknowns is equal to the rank of the augmented matrix.", "markdown": "A system of $m$ linear equations in $n$ unknowns is consistent if and only if the rank of the matrix of the coefficients of the unknowns is equal to the rank of the augmented matrix.", "why": "It gives the single test that tells a learner whether a system of equations has any solution at all.", "use": [ "lesson" ], "concepts": [ "concept/augmented-matrix", "concept/matrix", "theorem/consistency-of-a-linear-system" ] }, { "id": "dickson-theory-of-equations-1922/x-eb95b57f20", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "116", "location": "Determinants; Systems of Linear Equations", "latex": "For example, a determinant~$D$ of order~$3$ is of rank~$3$ if $D \\neq 0$; of rank~$2$ if $D = 0$, but some two-rowed minor is not zero; of rank~$1$ if every two-rowed minor is zero, but some element is not zero.", "markdown": "For example, a determinant $D$ of order $3$ is of rank $3$ if $D \\neq 0$; of rank $2$ if $D = 0$, but some two-rowed minor is not zero; of rank $1$ if every two-rowed minor is zero, but some element is not zero.", "why": "It walks through the three rank cases for a 3-by-3 determinant, which makes the abstract definition concrete.", "use": [ "lesson" ], "concepts": [ "concept/element-of-a-determinant", "concept/minor", "concept/rank-of-a-determinant" ] }, { "id": "dickson-theory-of-equations-1922/x-8a150dc650", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Determinants; Systems of Linear Equations", "latex": "A necessary and sufficient condition that $n$~linear homogeneous equations in $n$~unknowns shall have a set of solutions, other than the trivial one in which each unknown is zero, is that the determinant of the coefficients be zero.", "markdown": "A necessary and sufficient condition that $n$ linear homogeneous equations in $n$ unknowns shall have a set of solutions, other than the trivial one in which each unknown is zero, is that the determinant of the coefficients be zero.", "why": "It links a yes-or-no question about nontrivial solutions to a single computable number, the determinant.", "use": [ "lesson", "website" ], "concepts": [ "concept/determinant", "concept/homogeneous-linear-equations", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/x-ee75d546a5", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "122", "location": "Determinants; Systems of Linear Equations", "latex": "For $r = 1$, this development becomes the known expansion of $D$ according to the elements of the first column~(§90); here $M_1 = e_{11}$.", "markdown": "For $r = 1$, this development becomes the known expansion of $D$ according to the elements of the first column (§90); here $M_1 = e_{11}$.", "why": "It shows a learner that Laplace's development generalises the familiar expansion along one column.", "use": [ "lesson" ], "concepts": [ "concept/element-of-a-determinant", "method/laplace-s-development" ] }, { "id": "dickson-theory-of-equations-1922/x-f07466ded6", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Determinants; Systems of Linear Equations", "latex": "The theorem was discovered by induction in 1750 by G.~Cramer.", "markdown": "The theorem was discovered by induction in 1750 by G. Cramer.", "why": "It gives the historical origin of the rule, dating it and naming its discoverer as the chapter reports him.", "use": [ "history" ], "concepts": [ "person/gabriel-cramer", "theorem/cramer-s-rule" ] }, { "id": "dickson-theory-of-equations-1922/x-c7df95e232", "chapter": "dickson-theory-of-equations-1922/ch-ix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "128", "location": "Symmetric Functions", "latex": "A rational function of the independent variables $x_1, x_2, \\dotsc, x_n$ is said to be \\emph{symmetric} in them if it is unaltered by the interchange of any two of the variables.", "markdown": "A rational function of the independent variables $x_1, x_2, \\dotsc, x_n$ is said to be *symmetric* in them if it is unaltered by the interchange of any two of the variables.", "why": "It gives the defining property of a symmetric function in one sentence, which is the starting point for every later exercise.", "use": [ "lesson" ], "concepts": [ "concept/symmetric-function" ] }, { "id": "dickson-theory-of-equations-1922/x-cbdf831b14", "chapter": "dickson-theory-of-equations-1922/ch-ix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "128", "location": "Symmetric Functions", "latex": "In general, if $t$ is a rational function of $x_1, \\dotsc, x_n, \\Sigma t$ denotes the sum of $t$ and all of the distinct functions obtained from $t$ by permutations of the variables; such a $\\Sigma$-function (read \\emph{sigma function}) is symmetric in $x_1, \\dotsc, x_n$.", "markdown": "In general, if $t$ is a rational function of $x_1, \\dotsc, x_n, \\Sigma t$ denotes the sum of $t$ and all of the distinct functions obtained from $t$ by permutations of the variables; such a $\\Sigma$-function (read *sigma function*) is symmetric in $x_1, \\dotsc, x_n$.", "why": "It shows how the sigma notation builds a symmetric function from any rational function by summing over permutations.", "use": [ "lesson" ], "concepts": [ "concept/permutation", "concept/sigma-function", "concept/symmetric-function" ] }, { "id": "dickson-theory-of-equations-1922/x-f32340740c", "chapter": "dickson-theory-of-equations-1922/ch-ix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "128", "location": "Symmetric Functions", "latex": "In particular, $\\Sigma \\alpha = \\alpha + \\beta + \\gamma$, $\\Sigma \\alpha\\beta$, and $\\alpha\\beta\\gamma$ are called the three \\emph{elementary symmetric functions} of $\\alpha$, $\\beta$, $\\gamma$.", "markdown": "In particular, $\\Sigma \\alpha = \\alpha + \\beta + \\gamma$, $\\Sigma \\alpha\\beta$, and $\\alpha\\beta\\gamma$ are called the three *elementary symmetric functions* of $\\alpha$, $\\beta$, $\\gamma$.", "why": "It names the three elementary symmetric functions of three variables, which the learner will then meet in every cubic exercise.", "use": [ "lesson" ], "concepts": [ "concept/elementary-symmetric-function" ] }, { "id": "dickson-theory-of-equations-1922/x-89f475ee0a", "chapter": "dickson-theory-of-equations-1922/ch-ix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "129", "location": "Symmetric Functions", "latex": "$(\\Sigma \\alpha)^2 = \\Sigma \\alpha^2 + 2\\Sigma \\alpha\\beta$, whence $\\Sigma \\alpha^2 = p^2 - 2q$.", "markdown": "$(\\Sigma \\alpha)^2 = \\Sigma \\alpha^2 + 2\\Sigma \\alpha\\beta$, whence $\\Sigma \\alpha^2 = p^2 - 2q$.", "why": "It is a short worked identity that shows how sums of squares of roots are found from the coefficients without solving the equation.", "use": [ "lesson" ], "concepts": [ "concept/elementary-symmetric-function", "concept/root-of-an-equation", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/x-27eb0767d3", "chapter": "dickson-theory-of-equations-1922/ch-x", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "143", "location": "Elimination, Resultants And Discriminants", "latex": "We call $R$ the \\emph{resultant} (or \\emph{eliminant}) of the two equations.", "markdown": "We call $R$ the *resultant* (or *eliminant*) of the two equations.", "why": "Gives the learner the name and the defining role of the resultant as the test for a common root.", "use": [ "lesson", "website" ], "concepts": [ "concept/common-root", "concept/resultant" ] }, { "id": "dickson-theory-of-equations-1922/x-697ac048f9", "chapter": "dickson-theory-of-equations-1922/ch-x", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "143", "location": "Elimination, Resultants And Discriminants", "latex": "Methods of elimination which seem plausible often yield not $R$ itself, but the product of~$R$ by an extraneous function of the coefficients.", "markdown": "Methods of elimination which seem plausible often yield not $R$ itself, but the product of $R$ by an extraneous function of the coefficients.", "why": "Warns that a plausible elimination can silently return the resultant multiplied by an extra factor.", "use": [ "lesson" ], "concepts": [ "concept/extraneous-factor", "concept/resultant", "method/elimination" ] }, { "id": "dickson-theory-of-equations-1922/x-f7317aa938", "chapter": "dickson-theory-of-equations-1922/ch-x", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "145", "location": "Elimination, Resultants And Discriminants", "latex": "Multiply the first equation by $x$ and the second by $x^2$ and $x$ in turn.", "markdown": "Multiply the first equation by $x$ and the second by $x^2$ and $x$ in turn.", "why": "Shows the first move of Sylvester's method, turning two equations into a linear system in the powers of x.", "use": [ "lesson" ], "concepts": [ "concept/homogeneous-linear-equations", "method/sylvester-s-dialytic-method" ] }, { "id": "dickson-theory-of-equations-1922/x-7a46dec5b9", "chapter": "dickson-theory-of-equations-1922/ch-x", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "152", "location": "Elimination, Resultants And Discriminants", "latex": "Evidently $D$ is unaltered by the interchange of any two roots.", "markdown": "Evidently $D$ is unaltered by the interchange of any two roots.", "why": "Explains why the discriminant is a symmetric function of the roots and therefore expressible in the coefficients.", "use": [ "lesson" ], "concepts": [ "concept/discriminant", "concept/elementary-symmetric-function" ] }, { "id": "dickson-theory-of-equations-1922/x-e155ab5f83", "chapter": "dickson-theory-of-equations-1922/ch-x", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "151", "location": "Elimination, Resultants And Discriminants", "latex": "The student should employ only methods of elimination (such as those due to Sylvester, Euler, and Bézout) which have been proved to lead to the true resultant.", "markdown": "The student should employ only methods of elimination (such as those due to Sylvester, Euler, and Bézout) which have been proved to lead to the true resultant.", "why": "States the practical rule that only proven elimination methods should be trusted to give the true resultant.", "use": [ "lesson", "website" ], "concepts": [ "concept/resultant", "method/b-zout-s-method-of-elimination", "method/euler-s-method-of-elimination", "method/sylvester-s-dialytic-method" ] }, { "id": "dickson-theory-of-equations-1922/x-68501f81c6", "chapter": "dickson-theory-of-equations-1922/ch-x", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "145", "location": "Elimination, Resultants And Discriminants", "latex": "Given without proof by Sylvester, \\textit{Philosophical Magazine}, 1840, p.~132.", "markdown": "Given without proof by Sylvester, *Philosophical Magazine*, 1840, p. 132.", "why": "Credits the origin of the dialytic method to Sylvester in an 1840 publication, a historical anchor for the learner.", "use": [ "history" ], "concepts": [ "method/sylvester-s-dialytic-method", "person/james-joseph-sylvester" ] }, { "id": "dickson-theory-of-equations-1922/x-96c7d33a88", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "155", "location": "Appendix", "latex": "has a complex \\(real or imaginary\\) root.", "markdown": "has a complex real or imaginary root.", "why": "States the fundamental theorem in one line: every polynomial equation of degree n with complex coefficients has a complex root.", "use": [ "lesson" ], "concepts": [ "concept/root", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "dickson-theory-of-equations-1922/x-e7c12b3a70", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "This becomes intuitive geometrically. The portion of the graph of $y = F(x)$ which extends from its point with the abscissa~$2$ to its point with the abscissa~$3$ either has a lowest point or else has several equally low points, each lower than all the remaining points.", "markdown": "This becomes intuitive geometrically. The portion of the graph of $y = F(x)$ which extends from its point with the abscissa $2$ to its point with the abscissa $3$ either has a lowest point or else has several equally low points, each lower than all the remaining points.", "why": "Gives a picture a learner can see before the arithmetic: a continuous graph on a closed interval must reach a lowest point.", "use": [ "lesson", "website" ], "concepts": [ "concept/abscissa", "concept/graph-of-a-function", "concept/minimum" ] }, { "id": "dickson-theory-of-equations-1922/x-e82f9974dc", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "155", "location": "Appendix", "latex": "This simplified proof consists in showing that the two curves represented by $\\phi(x, y) = 0$ and $\\psi(x, y) = 0$ have at least one point $(x_1, y_1)$ in common, so that $z_1 = x_1 + iy_1$ is a root of $f(z)= 0$.", "markdown": "This simplified proof consists in showing that the two curves represented by $\\phi(x, y) = 0$ and $\\psi(x, y) = 0$ have at least one point $(x_1, y_1)$ in common, so that $z_1 = x_1 + iy_1$ is a root of $f(z)= 0$.", "why": "Explains the geometric meaning of a root: a point common to two curves in the plane gives a complex root.", "use": [ "lesson", "history" ], "concepts": [ "concept/complex-number", "concept/curve", "concept/root", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "dickson-theory-of-equations-1922/x-7d58af1318", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "158", "location": "Appendix", "latex": "In other words, if $|f(z)|\\leqq P$, the point representing~$z$ is inside circle~$C$.", "markdown": "In other words, if $|f(z)|\\leqq P$, the point representing $z$ is inside circle $C$.", "why": "Turns an inequality on |f(z)| into a statement about where points lie in the plane, a key step a learner can follow.", "use": [ "lesson" ], "concepts": [ "concept/circle", "concept/proof", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/x-da3afa5ffb", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "155", "location": "Appendix", "latex": "The proof differs from that of the auxiliary theorem in~§62 only in reading ``in absolute value'' for ``numerically.''", "markdown": "The proof differs from that of the auxiliary theorem in §62 only in reading “in absolute value” for “numerically.”", "why": "Shows how the older word 'numerically' became the modern 'absolute value' for complex numbers, a useful historical note on terminology.", "use": [ "history" ], "concepts": [ "concept/absolute-value" ] } ], "equations": [ { "id": "dickson-theory-of-equations-1922/eq-498dcea9b8", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "1", "location": "Complex Numbers", "latex": "i^2 = -1", "name": null, "statement": "The imaginary unit i is defined so that its square is minus one.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "i", "meaning": "imaginary unit, with i^2 = -1" } ], "sympy": "Eq(I**2, -1)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/imaginary-root" ] }, { "id": "dickson-theory-of-equations-1922/eq-ae3fd510eb", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "1", "location": "Complex Numbers", "latex": "(\\sqrt{p})^2 i^2 = -p", "name": null, "statement": "The square of either root of x^2 = -p, written as ±√p i, equals -p.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "positive real number" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((sqrt(p))**2*I**2, -p)", "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-a091be94e3", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Complex Numbers", "latex": "(a+bi) + (c+di) = (a+c) + (b+d)i", "name": null, "statement": "Two complex numbers are added by adding their real parts and their imaginary parts separately.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real part of the first complex number" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient of the first complex number" }, { "unit": null, "symbol": "c", "meaning": "real part of the second complex number" }, { "unit": null, "symbol": "d", "meaning": "imaginary coefficient of the second complex number" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((a+b*I)+(c+d*I), (a+c)+(b+d)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/sum", "method/addition" ] }, { "id": "dickson-theory-of-equations-1922/eq-0961b35b5e", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Complex Numbers", "latex": "(a+bi) - (c+di) = (a-c) + (b-d)i", "name": null, "statement": "Subtraction of complex numbers is defined by subtracting real parts and imaginary parts separately; it is the inverse of addition.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real part of the first complex number" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient of the first complex number" }, { "unit": null, "symbol": "c", "meaning": "real part of the second complex number" }, { "unit": null, "symbol": "d", "meaning": "imaginary coefficient of the second complex number" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((a+b*I)-(c+d*I), (a-c)+(b-d)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/difference", "method/subtraction" ] }, { "id": "dickson-theory-of-equations-1922/eq-0756b518ce", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Complex Numbers", "latex": "(a+bi)(c+di) = ac-bd+(ad+bc)i", "name": null, "statement": "Multiplication of complex numbers follows formal algebra, with i^2 replaced by -1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real part of the first complex number" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient of the first complex number" }, { "unit": null, "symbol": "c", "meaning": "real part of the second complex number" }, { "unit": null, "symbol": "d", "meaning": "imaginary coefficient of the second complex number" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((a+b*I)*(c+d*I), a*c-b*d+(a*d+b*c)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/product", "method/multiplication" ] }, { "id": "dickson-theory-of-equations-1922/eq-7d97384470", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Complex Numbers", "latex": "(a+bi)(a-bi) = a^2-b^2i^2 = a^2+b^2", "name": null, "statement": "A complex number times its conjugate equals the sum of the squares of its real and imaginary parts.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real part" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((a+b*I)*(a-b*I), a**2+b**2)", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/conjugate-complex-numbers", "method/multiplication" ] }, { "id": "dickson-theory-of-equations-1922/eq-728c6eae67", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Complex Numbers", "latex": "\\frac{e+fi}{a+bi} = \\frac{(e+fi)(a-bi)}{a^2+b^2} = \\frac{ae+bf}{a^2+b^2} + \\frac{af-be}{a^2+b^2} i", "name": null, "statement": "Division of complex numbers is performed by multiplying numerator and denominator by the conjugate of the denominator.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real part of the divisor" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient of the divisor" }, { "unit": null, "symbol": "e", "meaning": "real part of the dividend" }, { "unit": null, "symbol": "f", "meaning": "imaginary coefficient of the dividend" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((e+f*I)/(a+b*I), (a*e+b*f)/(a**2+b**2) + (a*f-b*e)/(a**2+b**2)*I)", "physics": false, "states": [], "concepts": [ "concept/conjugate-complex-numbers", "concept/quotient", "method/division" ] }, { "id": "dickson-theory-of-equations-1922/eq-4338bef33c", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Complex Numbers", "latex": "a^2+b^2 = 0", "name": null, "statement": "For real a and b, a^2+b^2 = 0 forces a = b = 0, so division by a nonzero complex number is always possible.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real number" }, { "unit": null, "symbol": "b", "meaning": "real number" } ], "sympy": "Eq(a**2+b**2, 0)", "physics": false, "states": [], "concepts": [ "concept/real-number", "concept/zero-displacement", "method/division" ] }, { "id": "dickson-theory-of-equations-1922/eq-c3515f4ce7", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "1", "location": "Complex Numbers", "latex": "a+bi=0", "name": null, "statement": "A complex number is zero if and only if both its real and imaginary parts are zero.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real part" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(a+b*I, 0)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/equality", "concept/zero-displacement" ] }, { "id": "dickson-theory-of-equations-1922/eq-7932989444", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "r = \\sqrt{a^2+b^2}", "name": null, "statement": "The modulus (absolute value) of a+bi is the positive length r of the segment from the origin to the point (a, b).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "r", "meaning": "modulus (absolute value) of a+bi, the length of OA" }, { "unit": null, "symbol": "a", "meaning": "real part" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient" } ], "sympy": "Eq(r, sqrt(a**2+b**2))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-d42a1beb66", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\cos \\theta = a/r", "name": null, "statement": "The amplitude θ of a+bi has cosine equal to the real part divided by the modulus.", "kind": "definition", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "amplitude (argument) of a+bi, the angle XOA" }, { "unit": null, "symbol": "a", "meaning": "real part" }, { "unit": null, "symbol": "r", "meaning": "modulus of a+bi" } ], "sympy": "Eq(cos(theta), a/r)", "physics": false, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/cosine", "quantity/amplitude-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-98a5aaf591", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\sin \\theta = b/r", "name": null, "statement": "The amplitude θ of a+bi has sine equal to the imaginary coefficient divided by the modulus.", "kind": "definition", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "amplitude (argument) of a+bi, the angle XOA" }, { "unit": null, "symbol": "b", "meaning": "imaginary coefficient" }, { "unit": null, "symbol": "r", "meaning": "modulus of a+bi" } ], "sympy": "Eq(sin(theta), b/r)", "physics": false, "states": [], "concepts": [ "concept/sine", "quantity/amplitude-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-238d308122", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "a+bi = r(\\cos\\theta + i\\sin\\theta)", "name": null, "statement": "A complex number equals its modulus times the cosine plus i times the sine of its amplitude (trigonometric form).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "r", "meaning": "modulus of a+bi" }, { "unit": "degree", "symbol": "θ", "meaning": "amplitude of a+bi" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(a+b*I, r*(cos(theta)+I*sin(theta)))", "physics": false, "states": [], "concepts": [ "concept/trigonometric-form-of-a-complex-number", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-f6de462048", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\omega = -\\tfrac{1}{2} + \\tfrac{1}{2}\\sqrt{3}i", "name": null, "statement": "The complex cube root of unity ω has real part -1/2 and imaginary part √3/2.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "ω", "meaning": "complex cube root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(omega, -Rational(1,2) + Rational(1,2)*sqrt(3)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cube-root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-bf5c9ba49e", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "x^3-1 = (x-1) (x^2+x+1)", "name": null, "statement": "The difference of cubes factors as (x-1) times (x^2+x+1), so the roots of x^3 = 1 are 1 and the roots of x^2+x+1 = 0.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (unknown complex number)" } ], "sympy": "Eq(x**3-1, (x-1)*(x**2+x+1))", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/factor", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-c0a9d3cd5a", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "(x + \\tfrac{1}{2})^2 = -\\tfrac{3}{4}", "name": null, "statement": "Completing the square in x^2+x+1 = 0 gives this relation, whose roots are the two complex cube roots of unity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (unknown complex number)" } ], "sympy": "Eq((x+Rational(1,2))**2, -Rational(3,4))", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-5e9b19b016", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\omega^2 + \\omega+1 = 0", "name": null, "statement": "The complex cube root of unity ω satisfies ω^2 + ω + 1 = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ω", "meaning": "complex cube root of unity" } ], "sympy": "Eq(omega**2+omega+1, 0)", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-dbefbba7cb", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\omega^3 = 1", "name": null, "statement": "The complex cube root of unity ω cubed equals one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ω", "meaning": "complex cube root of unity" } ], "sympy": "Eq(omega**3, 1)", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/power" ] }, { "id": "dickson-theory-of-equations-1922/eq-c5a6b9f2ee", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\omega \\omega' = 1", "name": null, "statement": "The two complex cube roots of unity ω and ω' have product one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ω", "meaning": "complex cube root of unity" }, { "unit": null, "symbol": "ω'", "meaning": "the other complex cube root of unity, the conjugate of ω" } ], "sympy": "Eq(omega*omega_prime, 1)", "physics": false, "states": [], "concepts": [ "concept/conjugate-complex-numbers", "concept/cube-root-of-unity", "concept/product" ] }, { "id": "dickson-theory-of-equations-1922/eq-f7cdff540f", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\omega' = \\omega^2", "name": null, "statement": "The second complex cube root of unity equals the square of ω.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ω'", "meaning": "the other complex cube root of unity" }, { "unit": null, "symbol": "ω", "meaning": "complex cube root of unity" } ], "sympy": "Eq(omega_prime, omega**2)", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/power" ] }, { "id": "dickson-theory-of-equations-1922/eq-8140068f41", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "3", "location": "Complex Numbers", "latex": "\\cos \\theta + i \\sin \\theta", "name": null, "statement": "placeholder", "kind": "definition", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [] }, { "id": "dickson-theory-of-equations-1922/eq-b19df795f5", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "4", "location": "Complex Numbers", "latex": "(\\cos \\theta + i \\sin \\theta) (\\cos \\alpha + i \\sin \\alpha) = \\cos (\\theta + \\alpha) + i \\sin (\\theta + \\alpha)", "name": null, "statement": "The product of two complex numbers in trigonometric form has modulus 1 and amplitude θ + α; with unit moduli this gives the addition formula for cosine and sine.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "amplitude of the first unit complex number" }, { "unit": "degree", "symbol": "α", "meaning": "amplitude of the second unit complex number" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((cos(theta)+I*sin(theta))*(cos(alpha)+I*sin(alpha)), cos(theta+alpha)+I*sin(theta+alpha))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "concept/trigonometric-form-of-a-complex-number", "theorem/modulus-and-amplitude-of-a-product" ] }, { "id": "dickson-theory-of-equations-1922/eq-7887749d4b", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "4", "location": "Complex Numbers", "latex": "\\frac{\\cos \\beta + i \\sin \\beta} {\\cos \\theta + i \\sin \\theta} = \\cos(\\beta - \\theta) + i \\sin(\\beta - \\theta)", "name": null, "statement": "The quotient of two complex numbers in trigonometric form has amplitude β - θ, the difference of the amplitudes.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "β", "meaning": "amplitude of the numerator" }, { "unit": "degree", "symbol": "θ", "meaning": "amplitude of the denominator" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((cos(beta)+I*sin(beta))/(cos(theta)+I*sin(theta)), cos(beta-theta)+I*sin(beta-theta))", "physics": false, "states": [], "concepts": [ "concept/quotient", "concept/trigonometric-form-of-a-complex-number", "quantity/amplitude-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-2fa905e224", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "5", "location": "Complex Numbers", "latex": "\\frac{1}{\\cos\\theta + i \\sin\\theta} = \\cos\\theta - i \\sin\\theta", "name": null, "statement": "The reciprocal of a unit complex number cos θ + i sin θ is its conjugate cos θ - i sin θ.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "amplitude of a unit complex number" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(1/(cos(theta)+I*sin(theta)), cos(theta)-I*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/conjugate-complex-numbers", "concept/quotient" ] }, { "id": "dickson-theory-of-equations-1922/eq-4e493c3b1b", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "5", "location": "Complex Numbers", "latex": "(\\cos\\theta + i \\sin\\theta)^n = \\cos n\\theta + i \\sin n\\theta", "name": "de Moivre's theorem", "statement": "For any positive whole number n, the nth power of cos θ + i sin θ equals cos nθ + i sin nθ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive whole number (exponent)" }, { "unit": "degree", "symbol": "θ", "meaning": "amplitude" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((cos(theta)+I*sin(theta))**n, cos(n*theta)+I*sin(n*theta))", "physics": false, "states": [ "theorem/de-moivre-s-theorem" ], "concepts": [ "concept/power", "concept/trigonometric-form-of-a-complex-number", "method/mathematical-induction" ] }, { "id": "dickson-theory-of-equations-1922/eq-9ac58b7b2a", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "5", "location": "Complex Numbers", "latex": "4\\sqrt{2} + 4\\sqrt{2} i = 8(\\cos 45° + i \\sin 45°)", "name": null, "statement": "The complex number 4√2 + 4√2 i has modulus 8 and amplitude 45° in trigonometric form.", "kind": "result", "symbols": [ { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(4*sqrt(2)+4*sqrt(2)*I, 8*(cos(pi/4)+I*sin(pi/4)))", "physics": false, "states": [], "concepts": [ "concept/trigonometric-form-of-a-complex-number", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-92e15a3d1c", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Complex Numbers", "latex": "3 \\theta = 45°+ k·360°", "name": null, "statement": "Equal cubes of complex numbers have amplitudes differing by a whole multiple of 360°, so 3θ = 45° + k·360° for an integer k.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "amplitude of a cube root" }, { "unit": null, "symbol": "k", "meaning": "integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integer", "concept/root", "quantity/amplitude-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-53ce9296a1", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Complex Numbers", "latex": "\\theta = 15°+k·120°", "name": null, "statement": "The amplitudes of the cube roots of 8(cos 45° + i sin 45°) are 15° + k·120° for integer k.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "amplitude of a cube root" }, { "unit": null, "symbol": "k", "meaning": "integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integer", "concept/root", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-8f53e62748", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "7", "location": "Complex Numbers", "latex": "r^n(\\cos n\\theta + i \\sin n\\theta) = \\cos A + i \\sin A", "name": null, "statement": "The nth power of r(cos θ + i sin θ), by de Moivre's theorem, must equal cos A + i sin A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "modulus of the root" }, { "unit": "degree", "symbol": "θ", "meaning": "amplitude of the root" }, { "unit": "degree", "symbol": "A", "meaning": "amplitude of the number whose nth root is sought" }, { "unit": null, "symbol": "n", "meaning": "positive whole number (order of the root)" } ], "sympy": "Eq(r**n*(cos(n*theta)+I*sin(n*theta)), cos(A)+I*sin(A))", "physics": false, "states": [], "concepts": [ "concept/root", "concept/trigonometric-form-of-a-complex-number", "theorem/de-moivre-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/eq-aaea467059", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "7", "location": "Complex Numbers", "latex": "n\\theta = A + k·360°", "name": null, "statement": "The amplitude of an nth root satisfies nθ = A + k·360° for an integer k.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "order of the root" }, { "unit": "degree", "symbol": "θ", "meaning": "amplitude of the root" }, { "unit": "degree", "symbol": "A", "meaning": "amplitude of the number" }, { "unit": null, "symbol": "k", "meaning": "integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integer", "concept/root", "quantity/amplitude-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-b88d06a9bb", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "7", "location": "Complex Numbers", "latex": "\\cos\\left(\\frac{A + k·360°}{n}\\right) + i \\sin\\left(\\frac{A + k·360°}{n}\\right)", "name": null, "statement": "The nth roots of cos A + i sin A are given by this expression for integer k; only k = 0, 1, ..., n-1 give distinct roots.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "A", "meaning": "amplitude of the number" }, { "unit": null, "symbol": "k", "meaning": "integer index of the root" }, { "unit": null, "symbol": "n", "meaning": "order of the root" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/root", "concept/root-of-unity", "quantity/amplitude-of-a-complex-number", "theorem/de-moivre-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/eq-09d00b99d9", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "8", "location": "Complex Numbers", "latex": "\\cos\\frac{2k \\pi}{n} + i \\sin\\frac{2k \\pi}{n}", "name": null, "statement": "The n distinct nth roots of unity are given by this expression for k = 0, 1, ..., n-1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "k", "meaning": "integer index, 0 to n-1" }, { "unit": null, "symbol": "n", "meaning": "order of the root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/eq-db6f90e194", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "8", "location": "Complex Numbers", "latex": "R = \\cos\\frac{2\\pi}{n} + i \\sin\\frac{2\\pi}{n}", "name": null, "statement": "R is the primitive nth root of unity with amplitude 2π/n radians, the case k = 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "primitive nth root of unity with amplitude 2π/n" }, { "unit": null, "symbol": "n", "meaning": "order of the root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(R, cos(2*pi/n)+I*sin(2*pi/n))", "physics": false, "states": [], "concepts": [ "concept/primitive-root-of-unity", "concept/root-of-unity", "quantity/pi", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/eq-74d3968fc7", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "8", "location": "Complex Numbers", "latex": "R,\\ R^2,\\ R^3,\\dotsc,\\ R^{n-1},\\ R^n = 1", "name": null, "statement": "The n distinct nth roots of unity are the powers of R, and the last power R^n equals 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "primitive nth root of unity" }, { "unit": null, "symbol": "n", "meaning": "order of the root of unity" } ], "sympy": "Eq(R**n, 1)", "physics": false, "states": [], "concepts": [ "concept/power", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-4c66e7b31b", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "8", "location": "Complex Numbers", "latex": "R = \\cos\\pi/2 + i \\sin\\pi/2 = i", "name": null, "statement": "For n = 4, R equals cos(π/2) + i sin(π/2), which is the imaginary unit i.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "primitive fourth root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(R, cos(pi/2)+I*sin(pi/2))", "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/root-of-unity", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/eq-57c8d2fa81", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Complex Numbers", "latex": "\\rho^n=1", "name": null, "statement": "A primitive nth root of unity ρ satisfies ρ^n = 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "ρ", "meaning": "primitive nth root of unity" }, { "unit": null, "symbol": "n", "meaning": "order of the root of unity" } ], "sympy": "Eq(rho**n, 1)", "physics": false, "states": [], "concepts": [ "concept/power", "concept/primitive-root-of-unity", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-504c7a4acc", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Complex Numbers", "latex": "\\rho^l \\neq 1", "name": null, "statement": "A primitive nth root of unity has no positive integral power l < n that equals 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "ρ", "meaning": "primitive nth root of unity" }, { "unit": null, "symbol": "l", "meaning": "positive integer less than n" }, { "unit": null, "symbol": "n", "meaning": "order of the root of unity" } ], "sympy": "Ne(rho**l, 1)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/power", "concept/primitive-root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-cf992cb854", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Complex Numbers", "latex": "(R^k)^{\\frac{n}{d}} = (R^n)^{\\frac{k}{d}} = 1", "name": null, "statement": "If k and n have a common divisor d > 1, then R^k is not a primitive nth root of unity, since its power n/d equals 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "primitive nth root of unity" }, { "unit": null, "symbol": "k", "meaning": "exponent" }, { "unit": null, "symbol": "n", "meaning": "order of the root of unity" }, { "unit": null, "symbol": "d", "meaning": "common divisor of k and n, d > 1" } ], "sympy": "Eq((R**k)**(n/d), 1)", "physics": false, "states": [], "concepts": [ "concept/divisor", "concept/power", "concept/primitive-root-of-unity", "concept/relatively-prime" ] }, { "id": "dickson-theory-of-equations-1922/eq-82fe1399d0", "chapter": "dickson-theory-of-equations-1922/ch-i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Complex Numbers", "latex": "R^{kl} = \\cos\\frac{2kl\\pi}{n} + i \\sin\\frac{2kl\\pi}{n}", "name": null, "statement": "By de Moivre's theorem, the power R^{kl} equals cos and sin of 2klπ/n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "primitive nth root of unity" }, { "unit": null, "symbol": "k", "meaning": "exponent" }, { "unit": null, "symbol": "l", "meaning": "positive integer exponent" }, { "unit": null, "symbol": "n", "meaning": "order of the root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(R**(k*l), cos(2*k*l*pi/n)+I*sin(2*k*l*pi/n))", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "theorem/de-moivre-s-theorem", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/eq-7d302667ca", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "ax^2 + bx + c = 0 \\quad (a \\ne 0)", "name": null, "statement": "The general quadratic equation, with a nonzero leading coefficient a, is called a quadratic equation or equation of the second degree.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "given number, the coefficient of x^2 (a ≠ 0)" }, { "unit": null, "symbol": "b", "meaning": "given number, the coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "given number, the constant term" }, { "unit": null, "symbol": "x", "meaning": "the variable (unknown)" } ], "sympy": "Eq(a*x**2 + b*x + c, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/quadratic-equation", "concept/unknown", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-82980b7e36", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "(2ax + b)^2 = \\Delta", "name": null, "statement": "Completing the square turns the quadratic into a square equal to the discriminant Delta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the quadratic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x in the quadratic" }, { "unit": null, "symbol": "x", "meaning": "the unknown" }, { "unit": null, "symbol": "Delta", "meaning": "discriminant b^2 - 4ac" } ], "sympy": "Eq((2*a*x + b)**2, Delta)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/quadratic-equation", "concept/unknown", "method/completing-the-square" ] }, { "id": "dickson-theory-of-equations-1922/eq-959a6e3ba9", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x_{1} = \\frac{-b + \\sqrt{\\Delta}}{2a}", "name": "quadratic formula", "statement": "The first root of the quadratic equation is given by the quadratic formula with the plus sign on the square root of the discriminant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "first root of the quadratic equation" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" }, { "unit": null, "symbol": "Delta", "meaning": "discriminant b^2 - 4ac" } ], "sympy": "Eq(x_1, (-b + sqrt(Delta))/(2*a))", "physics": false, "states": [ "method/quadratic-formula" ], "concepts": [ "concept/discriminant", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-42f482698a", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x_{2} = \\frac{-b - \\sqrt{\\Delta}}{2a}", "name": "quadratic formula", "statement": "The second root of the quadratic equation is given by the quadratic formula with the minus sign on the square root of the discriminant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_2", "meaning": "second root of the quadratic equation" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" }, { "unit": null, "symbol": "Delta", "meaning": "discriminant b^2 - 4ac" } ], "sympy": "Eq(x_2, (-b - sqrt(Delta))/(2*a))", "physics": false, "states": [ "method/quadratic-formula" ], "concepts": [ "concept/discriminant", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-65ac0615df", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x_{1} + x_{2} = \\frac{-b}{a}", "name": null, "statement": "The sum of the two roots of a quadratic equation is minus b over a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "first root" }, { "unit": null, "symbol": "x_2", "meaning": "second root" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" } ], "sympy": "Eq(x_1 + x_2, -b/a)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation", "concept/sum", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/eq-49606eed28", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x_{1} x_{2} = \\frac{ c}{a}", "name": null, "statement": "The product of the two roots of a quadratic equation is c over a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "first root" }, { "unit": null, "symbol": "x_2", "meaning": "second root" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(x_1*x_2, c/a)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/quadratic-equation", "concept/root-of-an-equation", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/eq-81959ac762", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "a(x - x_1)(x - x_2)\n \\equiv ax^2 - a(x_1 + x_2)x + ax_1 x_2\n \\equiv ax^2 + bx + c", "name": null, "statement": "For all x, the quadratic factors as a(x - x_1)(x - x_2), and this is identically equal to ax^2 + bx + c (the factored form).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "constant term" }, { "unit": null, "symbol": "x_1", "meaning": "first root" }, { "unit": null, "symbol": "x_2", "meaning": "second root" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq(a*(x - x_1)*(x - x_2), a*x**2 + b*x + c)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/factored-form", "concept/identity", "concept/linear-factor", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-d535ca0ab0", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "0 = ax_1^2 + bx_1 + c", "name": null, "statement": "Substituting the first root x_1 into the quadratic gives zero, so x_1 is a root of the equation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "first root" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(0, a*x_1**2 + b*x_1 + c)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-3d686061ea", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "0 = ax_2^2 + bx_2 + c", "name": null, "statement": "Substituting the second root x_2 into the quadratic gives zero, so x_2 is a root of the equation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_2", "meaning": "second root" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(0, a*x_2**2 + b*x_2 + c)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-24c3dbbbf9", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "11", "location": "Elementary Theorems on the Roots of an Equation", "latex": "\\Delta = b^2 - 4ac", "name": "discriminant", "statement": "The discriminant Delta of the quadratic function or equation is b^2 - 4ac.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "discriminant of the quadratic function ax^2 + bx + c" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(Delta, b**2 - 4*a*c)", "physics": false, "states": [ "concept/discriminant" ], "concepts": [ "concept/quadratic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-3dc51bfdb1", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "12", "location": "Elementary Theorems on the Roots of an Equation", "latex": "b^2 = 4ac", "name": null, "statement": "The quadratic ax^2 + bx + c is a perfect square of a linear function of x if and only if its discriminant is zero, b^2 = 4ac.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(b**2, 4*a*c)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/perfect-square", "concept/quadratic-function" ] }, { "id": "dickson-theory-of-equations-1922/eq-0828096f6e", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "12", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv c_0 x^n + c_1 x^{n-1} + \\dotsb + c_{n-1} x + c_n", "name": null, "statement": "A polynomial of degree n in x is a sum of constant multiples of successive powers of x, also called an integral rational function of degree n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "polynomial in x of degree n" }, { "unit": null, "symbol": "c_0, c_1, ..., c_n", "meaning": "constants (real or imaginary), with c_0 nonzero for degree n" }, { "unit": null, "symbol": "n", "meaning": "positive integer, the degree" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/constant", "concept/integer", "concept/integral", "concept/polynomial", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-62542de319", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "12", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv (x-c)q(x) + r", "name": "remainder theorem", "statement": "Dividing f(x) by x - c leaves a constant remainder r and quotient q(x), identically in x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "polynomial, the dividend" }, { "unit": null, "symbol": "c", "meaning": "number, the divisor is x - c" }, { "unit": null, "symbol": "q(x)", "meaning": "quotient polynomial" }, { "unit": null, "symbol": "r", "meaning": "constant remainder" } ], "sympy": "Eq(f(x), (x - c)*q(x) + r)", "physics": false, "states": [ "theorem/remainder-theorem" ], "concepts": [ "concept/dividend", "concept/divisor", "concept/identity", "concept/polynomial", "concept/quotient", "concept/remainder" ] }, { "id": "dickson-theory-of-equations-1922/eq-d92a6b31ec", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "12", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(c) = r", "name": "remainder theorem", "statement": "The remainder on dividing a polynomial f(x) by x - c equals f(c), the value of f at x = c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(c)", "meaning": "value of the polynomial when x = c" }, { "unit": null, "symbol": "r", "meaning": "remainder" }, { "unit": null, "symbol": "c", "meaning": "number" } ], "sympy": "Eq(f(c), r)", "physics": false, "states": [ "theorem/remainder-theorem" ], "concepts": [ "concept/remainder", "concept/value-of-a-function" ] }, { "id": "dickson-theory-of-equations-1922/eq-2f0fc55a5e", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "14", "location": "Elementary Theorems on the Roots of an Equation", "latex": "b_1 = a_1 + cb_0", "name": null, "statement": "In synthetic division by x - c, each new coefficient of the quotient is the next coefficient of f plus c times the previous quotient coefficient.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b_1", "meaning": "second coefficient of the quotient" }, { "unit": null, "symbol": "a_1", "meaning": "second coefficient of f(x)" }, { "unit": null, "symbol": "b_0", "meaning": "first coefficient of the quotient (equal to a_0)" }, { "unit": null, "symbol": "c", "meaning": "number, the divisor is x - c" } ], "sympy": "Eq(b_1, a_1 + c*b_0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/quotient-by-synthetic-division", "method/synthetic-division" ] }, { "id": "dickson-theory-of-equations-1922/eq-c492f00446", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "14", "location": "Elementary Theorems on the Roots of an Equation", "latex": "r = a_n + cb_{n-1}", "name": null, "statement": "In synthetic division by x - c, the remainder is the constant coefficient of f plus c times the last quotient coefficient.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "remainder" }, { "unit": null, "symbol": "a_n", "meaning": "constant term of f(x)" }, { "unit": null, "symbol": "b_{n-1}", "meaning": "last coefficient of the quotient" }, { "unit": null, "symbol": "c", "meaning": "number, the divisor is x - c" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant-term", "concept/remainder", "method/synthetic-division" ] }, { "id": "dickson-theory-of-equations-1922/eq-6b6d9c3e1c", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv (x - \\alpha_1)Q(x)", "name": "factor theorem", "statement": "If alpha_1 is a root of f(x) = 0, then x - alpha_1 is a factor, so f(x) equals (x - alpha_1) times a polynomial Q(x) of degree one less.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "polynomial of degree n" }, { "unit": null, "symbol": "alpha_1", "meaning": "a root of f(x) = 0 (may be complex)" }, { "unit": null, "symbol": "Q(x)", "meaning": "quotient polynomial of degree n-1" } ], "sympy": "Eq(f(x), (x - alpha1)*Q(x))", "physics": false, "states": [ "theorem/factor-theorem" ], "concepts": [ "concept/factor", "concept/factored-form", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-68ed424c59", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv c_0(x - \\alpha_1)(x - \\alpha_2) \\dotsm (x - \\alpha_n)", "name": null, "statement": "A polynomial of degree n with leading coefficient c_0 is a product of n linear factors built from its roots (the factored form).", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "polynomial of degree n" }, { "unit": null, "symbol": "c_0", "meaning": "leading coefficient, not zero" }, { "unit": null, "symbol": "alpha_1, ..., alpha_n", "meaning": "the roots of f(x) = 0 (may be complex), counted with multiplicity" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/factored-form", "concept/leading-coefficient", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-52bbe6e89b", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "16", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv c_0(x-\\alpha_1)^{m_1} (x-\\alpha_2)^{m_2} \\dotsm (x-\\alpha_k)^{m_k}, \\quad m_1 + m_2 + \\dotsb + m_k = n", "name": null, "statement": "With distinct roots alpha_1 to alpha_k of multiplicities m_1 to m_k, the polynomial factors with each linear factor raised to its multiplicity, and the multiplicities sum to the degree n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "alpha_j", "meaning": "distinct roots of f(x) = 0" }, { "unit": null, "symbol": "m_j", "meaning": "multiplicity of the root alpha_j" }, { "unit": null, "symbol": "k", "meaning": "number of distinct roots" }, { "unit": null, "symbol": "n", "meaning": "degree of f(x)" }, { "unit": null, "symbol": "c_0", "meaning": "leading coefficient" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/factored-form", "concept/multiple-root", "concept/root-of-an-equation", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-f1e0fc6d07", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Elementary Theorems on the Roots of an Equation", "latex": "a_0 = b_0", "name": null, "statement": "Two polynomials of degree n equal in value at more than n distinct points are term by term identical, so their leading coefficients agree (and likewise all coefficients).", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_0", "meaning": "leading coefficient of the first polynomial" }, { "unit": null, "symbol": "b_0", "meaning": "leading coefficient of the second polynomial" } ], "sympy": "Eq(a_0, b_0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/identity", "concept/number-of-roots", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-d1f8cc0481", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv c_0 x^n + c_1 x^{n-1} + \\dotsb + c_n = 0\\qquad (c_0 \\ne 0)", "name": null, "statement": "An equation f(x) = 0 of degree n with nonzero leading coefficient c_0 is the general polynomial equation of degree n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "polynomial of degree n" }, { "unit": null, "symbol": "c_0", "meaning": "leading coefficient, nonzero" }, { "unit": null, "symbol": "c_n", "meaning": "constant term" }, { "unit": null, "symbol": "n", "meaning": "degree" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/equation", "concept/polynomial", "concept/root-of-an-equation", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-e1a8a45424", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "18", "location": "Elementary Theorems on the Roots of an Equation", "latex": "(x - \\alpha_1)(x - \\alpha_2)\n &\\equiv x^2 - (\\alpha_1 + \\alpha_2)x + \\alpha_1\\alpha_2", "name": null, "statement": "The product of two linear factors expands to a quadratic whose x coefficient is minus the sum of the roots and whose constant is their product.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "alpha_1", "meaning": "first root" }, { "unit": null, "symbol": "alpha_2", "meaning": "second root" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq((x - alpha1)*(x - alpha2), x**2 - (alpha1 + alpha2)*x + alpha1*alpha2)", "physics": false, "states": [], "concepts": [ "concept/identity", "concept/linear-factor", "concept/product", "concept/sum", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/eq-841c5f4bce", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "18", "location": "Elementary Theorems on the Roots of an Equation", "latex": "(x - \\alpha_1)(x - \\alpha_2) \\dotsm (x - \\alpha_n)\n \\equiv x^n\n - (\\alpha_1 + \\dotsb + \\alpha_n)x^{n-1} \\\\\n + (\\alpha_1\\alpha_2 + \\alpha_1\\alpha_3 + \\alpha_2\\alpha_3 + \\dotsb\n + \\alpha_{n-1}\\alpha_n)x^{n-2} \\\\\n - (\\alpha_1\\alpha_2\\alpha_3 + \\alpha_1\\alpha_2\\alpha_4 + \\dotsb\n + \\alpha_{n-2}\\alpha_{n-1}\\alpha_n)x^{n-3} \\\\\n + \\dotsb + (-1)^n \\alpha_1\\alpha_2 \\dotsm \\alpha_n", "name": null, "statement": "The product of n linear factors expands with coefficients given by the elementary symmetric functions of the roots, with alternating signs; proved by mathematical induction.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "alpha_1, ..., alpha_n", "meaning": "the n roots" }, { "unit": null, "symbol": "n", "meaning": "number of linear factors" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/identity", "concept/linear-factor", "concept/product", "concept/sum", "method/mathematical-induction", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/eq-72621e8045", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "18", "location": "Elementary Theorems on the Roots of an Equation", "latex": "\\alpha_1 + \\alpha_2 + \\dotsb + \\alpha_n &= -c_1 / c_0", "name": null, "statement": "The sum of the roots of the equation equals minus c_1 over c_0, the negative of the coefficient of the second term over the leading coefficient.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha_i", "meaning": "the n roots of f(x) = 0" }, { "unit": null, "symbol": "c_0", "meaning": "leading coefficient" }, { "unit": null, "symbol": "c_1", "meaning": "coefficient of x^{n-1}" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/root-of-an-equation", "concept/sum", "concept/sum-of-the-roots", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/eq-5c36b0cdd4", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "18", "location": "Elementary Theorems on the Roots of an Equation", "latex": "\\alpha_1\\alpha_2 \\dotsm \\alpha_{n-1}\\alpha_n &= (-1)^n c_n / c_0", "name": null, "statement": "The product of all the roots equals (-1)^n times c_n over c_0, the constant term over the leading coefficient with sign.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha_i", "meaning": "the n roots of f(x) = 0" }, { "unit": null, "symbol": "n", "meaning": "degree" }, { "unit": null, "symbol": "c_0", "meaning": "leading coefficient" }, { "unit": null, "symbol": "c_n", "meaning": "constant term" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant-term", "concept/product", "concept/product-of-roots", "concept/root-of-an-equation", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/eq-61b7702517", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Elementary Theorems on the Roots of an Equation", "latex": "(x-a)^2 + b^2 \\equiv (x - a-bi)(x - a+bi)", "name": null, "statement": "The real quadratic (x - a)^2 + b^2 factors into the two conjugate complex linear factors x - (a + bi) and x - (a - bi).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real number" }, { "unit": null, "symbol": "b", "meaning": "real number, nonzero in the use made of it" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit, square root of -1" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq((x - a)**2 + b**2, (x - a - b*I)*(x - a + b*I))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/conjugate-complex-numbers", "concept/identity", "concept/imaginary-root", "concept/linear-factor" ] }, { "id": "dickson-theory-of-equations-1922/eq-1a457adaa8", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv Q(x)\\bigl\\{(x-a)^2 + b^2\\bigr\\} + rx + s", "name": null, "statement": "Dividing a real polynomial by (x - a)^2 + b^2 leaves a quotient Q(x) and a linear remainder rx + s, identically in x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "real polynomial" }, { "unit": null, "symbol": "Q(x)", "meaning": "quotient, real polynomial" }, { "unit": null, "symbol": "r", "meaning": "real coefficient of the remainder" }, { "unit": null, "symbol": "s", "meaning": "real constant of the remainder" }, { "unit": null, "symbol": "a", "meaning": "real number" }, { "unit": null, "symbol": "b", "meaning": "real number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/dividend", "concept/divisor", "concept/polynomial", "concept/quotient", "concept/real-number", "concept/remainder" ] }, { "id": "dickson-theory-of-equations-1922/eq-ed0dc2a88e", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "21", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x \\geqq 1 + \\sqrt[k]{G / a_0}", "name": "upper limit to the real roots (Theorem I)", "statement": "Any real x at least 1 plus the k-th root of G over a_0 is not a root, so that bound is an upper limit to the real roots.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a positive real number" }, { "unit": null, "symbol": "a_0", "meaning": "leading coefficient, positive" }, { "unit": null, "symbol": "G", "meaning": "greatest numerical value of the negative coefficients" }, { "unit": null, "symbol": "k", "meaning": "number of nonnegative coefficients preceding the first negative coefficient" } ], "sympy": null, "physics": false, "states": [ "theorem/upper-limit-to-the-real-roots-theorem-i" ], "concepts": [ "concept/coefficient", "concept/inequality", "concept/real-root", "concept/upper-limit-to-the-roots" ] }, { "id": "dickson-theory-of-equations-1922/eq-fcd915bb34", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "21", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x^{n-k} + \\dotsb + x + 1 \\equiv \\frac{x^{n-k+1} - 1}{x - 1}", "name": null, "statement": "The geometric sum of successive powers of x from 0 to n-k equals (x^{n-k+1} - 1) over (x - 1), for x not equal to 1.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable, not equal to 1" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" }, { "unit": null, "symbol": "k", "meaning": "index of the first negative coefficient" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/geometrical-progression", "concept/identity", "concept/power", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-63afa8a92e", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "22", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x^4 \\equiv (x-1) (x^3 + x^2 + x + 1) + 1", "name": null, "statement": "The fourth power of x is written as (x - 1) times a cubic in x plus 1, an identity used in the proof of Theorem II.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq(x**4, (x - 1)*(x**3 + x**2 + x + 1) + 1)", "physics": false, "states": [], "concepts": [ "concept/identity", "concept/polynomial", "concept/power", "concept/remainder" ] }, { "id": "dickson-theory-of-equations-1922/eq-c463c3d0b2", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "22", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x^2 \\equiv (x-1) (x+1) + 1", "name": null, "statement": "The square of x is written as (x - 1)(x + 1) plus 1, an identity used in the proof of Theorem II.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq(x**2, (x - 1)*(x + 1) + 1)", "physics": false, "states": [], "concepts": [ "concept/identity", "concept/power", "concept/square" ] }, { "id": "dickson-theory-of-equations-1922/eq-50dc27ba42", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x \\geqq 1 + \\frac{-a_{k_i}}{\\sum a_m}", "name": "upper limit to the roots (Theorem II)", "statement": "For each negative coefficient a_{k_i}, any x at least 1 plus minus a_{k_i} over the sum of the preceding positive coefficients is not a root; the greatest such bound is an upper limit to the roots.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a real number greater than 1" }, { "unit": null, "symbol": "a_{k_i}", "meaning": "a negative coefficient of the equation" }, { "unit": null, "symbol": "a_m", "meaning": "the positive coefficients preceding a_{k_i}" } ], "sympy": null, "physics": false, "states": [ "theorem/upper-limit-to-the-roots-theorem-ii" ], "concepts": [ "concept/coefficient", "concept/inequality", "concept/positive-number", "concept/sum", "concept/upper-limit-to-the-roots" ] }, { "id": "dickson-theory-of-equations-1922/eq-3141860cb9", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Elementary Theorems on the Roots of an Equation", "latex": "x \\geqq 1 + \\frac{-a_0}{\\sum a_m}", "name": "upper limit to the roots (Theorem II)", "statement": "When the constant term a_0 is negative, any x at least 1 plus minus a_0 over the sum of the positive coefficients is not a root.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a real number" }, { "unit": null, "symbol": "a_0", "meaning": "constant term, negative in this case" }, { "unit": null, "symbol": "a_m", "meaning": "the positive coefficients" } ], "sympy": null, "physics": false, "states": [ "theorem/upper-limit-to-the-roots-theorem-ii" ], "concepts": [ "concept/constant-term", "concept/inequality", "concept/positive-number", "concept/sum", "concept/upper-limit-to-the-roots" ] }, { "id": "dickson-theory-of-equations-1922/eq-b7dd6bc71a", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "24", "location": "Elementary Theorems on the Roots of an Equation", "latex": "a_0 x^n + \\dotsb + a_{n-1}x + a_n = 0", "name": null, "statement": "A polynomial equation of degree n in the unknown x, whose coefficients a_0, ..., a_n are integers, is set equal to zero; this is the equation the integral-root theorem is applied to.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a_0, ..., a_n", "meaning": "coefficients of the equation (integers in this context)" }, { "unit": null, "symbol": "x", "meaning": "unknown (the root being sought)" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation (highest power of the unknown)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant-term", "concept/equation", "concept/integer", "concept/polynomial", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-2896e2d772", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "25", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv x^4 - 9x^3 + 24x^2 - 23x + 15 = 0", "name": null, "statement": "The polynomial f(x) is defined as the quartic x^4 - 9x^3 + 24x^2 - 23x + 15, and the equation f(x) = 0 is the one whose roots are sought.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "the polynomial in x under study" }, { "unit": null, "symbol": "x", "meaning": "unknown" } ], "sympy": "Eq(f(x), x**4 - 9*x**3 + 24*x**2 - 23*x + 15)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant-term", "concept/function", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-744aeec469", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "25", "location": "Elementary Theorems on the Roots of an Equation", "latex": "15y^4 - 23y^3 + 24y^2 - 9y + 1 = 0", "name": null, "statement": "Replacing x by 1/y in f(x) = 0 and multiplying through by y^4 gives this equation in y, the transformed equation whose root y = 1/3 corresponds to the root x = 3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "new unknown, equal to 1/x" } ], "sympy": "Eq(15*y**4 - 23*y**3 + 24*y**2 - 9*y + 1, 0)", "physics": false, "states": [], "concepts": [ "concept/reciprocal", "concept/root-of-an-equation", "concept/transformed-equation", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-f90c60584a", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "27", "location": "Elementary Theorems on the Roots of an Equation", "latex": "c_0 x^n + c_1 x^{n-1} + \\dotsb + c_{n-1} x + c_n = 0", "name": null, "statement": "The general equation of degree n with integral coefficients c_0, ..., c_n, to which the rational-root theorem applies.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c_0, ..., c_n", "meaning": "coefficients of the equation (integers)" }, { "unit": null, "symbol": "x", "meaning": "unknown" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/equation", "concept/integer", "concept/polynomial", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-175d468f2c", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "26", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(x) \\equiv (x-d)Q(x)", "name": null, "statement": "If d is a root of f(x) = 0, then f(x) is identically the product of (x - d) and a polynomial Q(x) with integral coefficients.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "the polynomial under study" }, { "unit": null, "symbol": "d", "meaning": "a root of f(x) = 0 (an integral divisor of the constant term)" }, { "unit": null, "symbol": "Q(x)", "meaning": "quotient polynomial with integral coefficients" } ], "sympy": "Eq(f(x), (x - d)*Q(x))", "physics": false, "states": [], "concepts": [ "concept/divisor", "concept/factor", "concept/integer", "concept/polynomial", "concept/quotient", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-fdeb715f60", "chapter": "dickson-theory-of-equations-1922/ch-ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "26", "location": "Elementary Theorems on the Roots of an Equation", "latex": "f(m) = (m-d) q", "name": null, "statement": "For an integer m, the value f(m) is divisible by m - d whenever d is a root, because q = Q(m) is an integer; this gives the test used to exclude candidate roots.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(m)", "meaning": "value of the polynomial at the chosen integer m" }, { "unit": null, "symbol": "m", "meaning": "any chosen integer" }, { "unit": null, "symbol": "d", "meaning": "an integral divisor of the constant term being tested as a root" }, { "unit": null, "symbol": "q", "meaning": "the integer Q(m)" } ], "sympy": "Eq(f(m), (m - d)*q)", "physics": false, "states": [], "concepts": [ "concept/divisibility", "concept/divisor", "concept/integer", "concept/root-of-an-equation", "concept/value-of-a-function" ] }, { "id": "dickson-theory-of-equations-1922/eq-7ec82acf8c", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "29", "location": "Constructions with Ruler and Compasses", "latex": "x^2 - ax + b = 0", "name": null, "statement": "The quadratic whose real roots are constructed as the abscissas where a circle cuts the x-axis.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown, the root being sought" }, { "unit": null, "symbol": "a", "meaning": "given constructible coefficient of x" }, { "unit": null, "symbol": "b", "meaning": "given constructible constant term" } ], "sympy": "Eq(x**2 - a*x + b, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/construction", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-051f7a6f18", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "29", "location": "Constructions with Ruler and Compasses", "latex": "\\left(x - \\frac{a}{2}\\right)^2 + \\left(y - \\frac{b+1}{2}\\right)^2 = \\frac{a^2 + (b-1)^2}{4}", "name": null, "statement": "Equation of the circle having the segment BQ, with B=(0,1) and Q=(a,b), as a diameter.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a general point on the circle" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a general point on the circle" }, { "unit": null, "symbol": "a", "meaning": "abscissa of the point Q" }, { "unit": null, "symbol": "b", "meaning": "ordinate of the point Q" } ], "sympy": "Eq((x - a/2)**2 + (y - (b+1)/2)**2, (a**2 + (b-1)**2)/4)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/centre-of-a-circle", "concept/circle", "concept/diameter" ] }, { "id": "dickson-theory-of-equations-1922/eq-c0b1cf9c5b", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "31", "location": "Constructions with Ruler and Compasses", "latex": "y = mx + b", "name": null, "statement": "Equation of a straight line in coordinates, with slope m and intercept b.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate of a general point on the line" }, { "unit": null, "symbol": "x", "meaning": "abscissa of a general point on the line" }, { "unit": null, "symbol": "m", "meaning": "slope of the line" }, { "unit": null, "symbol": "b", "meaning": "intercept of the line on the y-axis" } ], "sympy": "Eq(y, m*x + b)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/line", "concept/linear-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-6fac5553c3", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "31", "location": "Constructions with Ruler and Compasses", "latex": "x = \\frac{b' - b}{m - m'}", "name": null, "statement": "Abscissa of the intersection of the lines y = mx + b and y = m'x + b'; a rational function of the coefficients.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of the point of intersection" }, { "unit": null, "symbol": "m, m'", "meaning": "slopes of the two lines" }, { "unit": null, "symbol": "b, b'", "meaning": "intercepts of the two lines" } ], "sympy": "Eq(x, (bp - b)/(m - mp))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/line", "concept/point", "concept/rational-function" ] }, { "id": "dickson-theory-of-equations-1922/eq-eb38515661", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "31", "location": "Constructions with Ruler and Compasses", "latex": "y = \\frac{mb' - m'b}{m - m'}", "name": null, "statement": "Ordinate of the intersection of the two lines y = mx + b and y = m'x + b'; a rational function of the coefficients.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate of the point of intersection" }, { "unit": null, "symbol": "m, m'", "meaning": "slopes of the two lines" }, { "unit": null, "symbol": "b, b'", "meaning": "intercepts of the two lines" } ], "sympy": "Eq(y, (m*bp - mp*b)/(m - mp))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/line", "concept/point", "concept/rational-function" ] }, { "id": "dickson-theory-of-equations-1922/eq-7ec99d9584", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "31", "location": "Constructions with Ruler and Compasses", "latex": "(x - c)^2 + (y - d)^2 = r^2", "name": null, "statement": "Equation of a circle with centre (c, d) and radius r.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c, d", "meaning": "coordinates of the centre of the circle" }, { "unit": "unit length", "symbol": "r", "meaning": "radius of the circle" }, { "unit": null, "symbol": "x", "meaning": "abscissa of a general point on the circle" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a general point on the circle" } ], "sympy": "Eq((x - c)**2 + (y - d)**2, r**2)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/centre-of-a-circle", "concept/circle", "concept/radius-of-a-regular-polygon" ] }, { "id": "dickson-theory-of-equations-1922/eq-fcfbab8685", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "30", "location": "Constructions with Ruler and Compasses", "latex": "\\tfrac{1}{2}(a ± \\sqrt{a^2 -4b)}", "name": null, "statement": "Lengths of the segments constructed in the Remark after the criterion, the two roots of x^2 - ax + b = 0 halved-sum form. The printed brace is unbalanced in the source (sqrt{a^2 -4b) ), a typesetting point left as printed; the intended form is (a ± sqrt(a^2 - 4b))/2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b", "meaning": "given constructible numbers of the quadratic x^2 - ax + b = 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-1042fbdf3f", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "32", "location": "Constructions with Ruler and Compasses", "latex": "s = \\tfrac{1}{2}\\sqrt{10 - 2\\sqrt{5}}", "name": null, "statement": "Side of a regular pentagon inscribed in a circle of unit radius, a number constructible by ruler and compasses.", "kind": "result", "symbols": [ { "unit": "unit length", "symbol": "s", "meaning": "side of a regular pentagon inscribed in a circle of radius unity" } ], "sympy": "Eq(s, sqrt(10 - 2*sqrt(5))/2)", "physics": false, "states": [], "concepts": [ "concept/construction", "concept/pentagon", "concept/radius-of-a-regular-polygon", "concept/regular-polygon" ] }, { "id": "dickson-theory-of-equations-1922/eq-3734d44d89", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "32", "location": "Constructions with Ruler and Compasses", "latex": "x^3 + \\alpha x^2 + \\beta x + \\gamma = 0", "name": null, "statement": "General cubic equation with rational coefficients, whose roots are examined for constructibility.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown, the root sought" }, { "unit": null, "symbol": "α, β, γ", "meaning": "rational coefficients of the cubic" } ], "sympy": "Eq(x**3 + alpha*x**2 + beta*x + gamma, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/cubic-equation", "concept/rational-number", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-7035756e71", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "33", "location": "Constructions with Ruler and Compasses", "latex": "x_1 = \\frac{a + b \\sqrt{k}}{c + d \\sqrt{k}}", "name": null, "statement": "A root of a superimposed-radical expression written as a fraction with the highest-order radical sqrt(k) in numerator and denominator.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a constructible root of the cubic" }, { "unit": null, "symbol": "a, b, c, d", "meaning": "quantities not involving sqrt(k)" }, { "unit": null, "symbol": "k", "meaning": "rational number under the highest-order radical" } ], "sympy": "Eq(x1, (a + b*sqrt(k))/(c + d*sqrt(k)))", "physics": false, "states": [], "concepts": [ "concept/radical-sign", "concept/rational-expression", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-ec9001ece3", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "33", "location": "Constructions with Ruler and Compasses", "latex": "x_1 = e + f\\sqrt{k}", "name": null, "statement": "The root x_1 written as e + f·sqrt(k), with e and f free of sqrt(k).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "the constructible root of the cubic" }, { "unit": null, "symbol": "e, f", "meaning": "quantities not involving sqrt(k)" }, { "unit": null, "symbol": "k", "meaning": "rational number under the highest-order radical" } ], "sympy": "Eq(x1, e + f*sqrt(k))", "physics": false, "states": [], "concepts": [ "concept/radical-sign", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-cd2d133997", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "33", "location": "Constructions with Ruler and Compasses", "latex": "(e + f \\sqrt{k})^3 + \\alpha(e + f \\sqrt{k})^2 + \\beta(e + f \\sqrt{k}) + \\gamma = A + B\\sqrt{k}", "name": null, "statement": "Substituting x = e + f·sqrt(k) into the cubic and reducing gives A + B·sqrt(k), with A and B polynomials not involving sqrt(k).", "kind": "result", "symbols": [ { "unit": null, "symbol": "A, B", "meaning": "polynomials in e, f, k and the rational coefficients" }, { "unit": null, "symbol": "e, f", "meaning": "quantities in the root x_1" }, { "unit": null, "symbol": "k", "meaning": "rational number under the radical" } ], "sympy": "Eq((e + f*sqrt(k))**3 + alpha*(e + f*sqrt(k))**2 + beta*(e + f*sqrt(k)) + gamma, A + B*sqrt(k))", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/polynomial", "concept/radical-sign", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-b508460d24", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "34", "location": "Constructions with Ruler and Compasses", "latex": "x_2 = e - f \\sqrt{k}", "name": null, "statement": "The conjugate e - f·sqrt(k) is a second root of the cubic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_2", "meaning": "second root of the cubic" }, { "unit": null, "symbol": "e, f", "meaning": "quantities in x_1" }, { "unit": null, "symbol": "k", "meaning": "rational number under the radical" } ], "sympy": "Eq(x2, e - f*sqrt(k))", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/radical-sign", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-a977661c4a", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "34", "location": "Constructions with Ruler and Compasses", "latex": "x_3 = -\\alpha - x_1 - x_2 = -\\alpha - 2e", "name": null, "statement": "The third root equals minus alpha minus the sum of the other two roots, which is -alpha - 2e (sum of roots equals -alpha).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_3", "meaning": "third root of the cubic" }, { "unit": null, "symbol": "α", "meaning": "rational coefficient of x^2 in the cubic" }, { "unit": null, "symbol": "e", "meaning": "rational part of x_1 under consideration" } ], "sympy": "Eq(x3, -alpha - 2*e)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-2b55e8aed3", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "34", "location": "Constructions with Ruler and Compasses", "latex": "x_3 = g + h \\sqrt{s}", "name": null, "statement": "If e is irrational, the third root x_3 takes the form g + h·sqrt(s) with h not zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_3", "meaning": "third root of the cubic" }, { "unit": null, "symbol": "g, h", "meaning": "quantities not involving sqrt(s)" }, { "unit": null, "symbol": "s", "meaning": "rational number under the highest-order radical of e" } ], "sympy": "Eq(x3, g + h*sqrt(s))", "physics": false, "states": [], "concepts": [ "concept/radical-sign", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-4ee77a2fcc", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "34", "location": "Constructions with Ruler and Compasses", "latex": "g - h \\sqrt{s} = e ± f \\sqrt{k}", "name": null, "statement": "The conjugate g - h·sqrt(s) of x_3 is one of the other two roots, so it equals e ± f·sqrt(k); the sign ± is not resolved by the relation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "g, h", "meaning": "quantities in x_3" }, { "unit": null, "symbol": "s", "meaning": "radical in e" }, { "unit": null, "symbol": "e, f, k", "meaning": "quantities of x_1 and x_2" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/radical-sign", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-26c685639e", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "34", "location": "Constructions with Ruler and Compasses", "latex": "\\cos A = 4 \\cos^3 \\frac{A}{3} - 3 \\cos \\frac{A}{3}", "name": "triple-angle formula for cosine", "statement": "The cosine of an angle expressed through the cosine of one third of it.", "kind": "identity", "symbols": [ { "unit": "degree of angle", "symbol": "A", "meaning": "a given angle" } ], "sympy": "Eq(cos(A), 4*cos(A/3)**3 - 3*cos(A/3))", "physics": false, "states": [ "theorem/triple-angle-formula-for-cosine" ], "concepts": [ "concept/cosine", "concept/trisection-of-an-angle", "quantity/angle", "theorem/trigonometric-identity" ] }, { "id": "dickson-theory-of-equations-1922/eq-6492bf3ef7", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "x^3 - 3x = 2\\cos A", "name": null, "statement": "With x = 2 cos(A/3), the angle A gives the cubic x^3 - 3x = 2 cos A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "2 cos(A/3)" }, { "unit": "degree of angle", "symbol": "A", "meaning": "a given angle" } ], "sympy": "Eq(x**3 - 3*x, 2*cos(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cubic-equation", "concept/trisection-of-an-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-b79aff2c45", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "\\cos A = -\\frac{1}{2}", "name": null, "statement": "For the angle A = 120 degrees, the cosine of A is -1/2.", "kind": "result", "symbols": [ { "unit": "degree of angle", "symbol": "A", "meaning": "the angle 120 degrees" } ], "sympy": "Eq(cos(A), Rational(-1, 2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/trisection-of-an-angle", "quantity/angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-c220d80036", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "x^3 - 3x + 1 = 0", "name": null, "statement": "The cubic whose root 2 cos(A/3) gives the trisection of A = 120 degrees; it has no rational root.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "2 cos(A/3), with A = 120 degrees" } ], "sympy": "Eq(x**3 - 3*x + 1, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/rational-root", "concept/trisection-of-an-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-5e538f39c6", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "x = 2 \\cos B", "name": null, "statement": "Defines x as twice the cosine of an angle B that would be constructible by ruler and compasses.", "kind": "definition", "symbols": [ { "unit": "unit length", "symbol": "x", "meaning": "twice the cosine of the angle B" }, { "unit": "degree of angle", "symbol": "B", "meaning": "angle containing 360/7 degrees" } ], "sympy": "Eq(x, 2*cos(B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/regular-polygon", "quantity/angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-67e891a912", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "\\cos 3B = \\cos 4B", "name": null, "statement": "Since 7B = 360 degrees, the angles 3B and 4B have equal cosines.", "kind": "result", "symbols": [ { "unit": "degree of angle", "symbol": "B", "meaning": "angle of 360/7 degrees" } ], "sympy": "Eq(cos(3*B), cos(4*B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/regular-polygon", "quantity/angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-5c388569d0", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "2(4 \\cos^3 B - 3 \\cos B) = x^3 - 3x", "name": null, "statement": "Twice the cosine of 3B, expressed in x = 2 cos B, equals x^3 - 3x.", "kind": "identity", "symbols": [ { "unit": "degree of angle", "symbol": "B", "meaning": "angle of 360/7 degrees" }, { "unit": null, "symbol": "x", "meaning": "2 cos B" } ], "sympy": "Eq(2*(4*cos(B)**3 - 3*cos(B)), x**3 - 3*x)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/polynomial", "theorem/trigonometric-identity" ] }, { "id": "dickson-theory-of-equations-1922/eq-b01e00317d", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "4(2 \\cos^2 B - 1)^2 - 2 = (x^2 - 2)^2 - 2", "name": null, "statement": "Twice the cosine of 4B, expressed in x = 2 cos B, equals (x^2 - 2)^2 - 2.", "kind": "identity", "symbols": [ { "unit": "degree of angle", "symbol": "B", "meaning": "angle of 360/7 degrees" }, { "unit": null, "symbol": "x", "meaning": "2 cos B" } ], "sympy": "Eq(4*(2*cos(B)**2 - 1)**2 - 2, (x**2 - 2)**2 - 2)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/polynomial", "theorem/trigonometric-identity" ] }, { "id": "dickson-theory-of-equations-1922/eq-70f9c72cd5", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "0 = x^4 - 4x^2 + 2 - (x^3 - 3x) = (x - 2)(x^3 + x^2 - 2x - 1)", "name": null, "statement": "Equating the two expressions for 2 cos 3B and 2 cos 4B gives a quartic which factors as (x - 2)(x^3 + x^2 - 2x - 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "2 cos B" } ], "sympy": "Eq(0, x**4 - 4*x**2 + 2 - (x**3 - 3*x))", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/factor", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-e329073739", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "35", "location": "Constructions with Ruler and Compasses", "latex": "x^3 + x^2 - 2x - 1 = 0", "name": null, "statement": "The cubic satisfied by x = 2 cos B for the regular polygon of 7 sides; it has no rational root, so the heptagon cannot be constructed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "2 cos B, twice the cosine of 360/7 degrees" } ], "sympy": "Eq(x**3 + x**2 - 2*x - 1, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/rational-root", "concept/regular-polygon", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-c5827ece03", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "R = \\cos\\frac{2\\pi}{7} + i \\sin\\frac{2\\pi}{7}", "name": null, "statement": "R is the complex seventh root of unity with argument 2 pi / 7.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "a primitive seventh root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(R, cos(2*pi/7) + I*sin(2*pi/7))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/regular-polygon", "concept/root-of-an-equation", "concept/trigonometric-form-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-5481ae5491", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "\\frac{1}{R} = \\cos\\frac{2\\pi}{7} -i \\sin\\frac{2\\pi}{7}", "name": null, "statement": "The reciprocal of R is its complex conjugate.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "R", "meaning": "the seventh root of unity defined above" } ], "sympy": "Eq(1/R, cos(2*pi/7) - I*sin(2*pi/7))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-ff10398641", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "R + \\frac{1}{R} = 2\\cos\\frac{2\\pi}{7}", "name": null, "statement": "The sum of R and its reciprocal is twice the cosine of 2 pi / 7.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "R", "meaning": "the seventh root of unity" } ], "sympy": "Eq(R + 1/R, 2*cos(2*pi/7))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cosine", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-7ded9bbf7a", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "y^6 + y^5 + y^4 + y^3 + y^2 + y + 1 = 0", "name": null, "statement": "The equation with roots R, R^2, ..., R^6, obtained by removing the factor y - 1 from y^7 - 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown, a seventh root of unity other than 1" } ], "sympy": "Eq(y**6 + y**5 + y**4 + y**3 + y**2 + y + 1, 0)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/factor", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-012ec9707c", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "y+ \\frac{1}{y} = x", "name": null, "statement": "The substitution that converts the reciprocal equation in y into an equation in x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the variable of the reciprocal equation" }, { "unit": null, "symbol": "x", "meaning": "new variable y + 1/y" } ], "sympy": "Eq(y + 1/y, x)", "physics": false, "states": [], "concepts": [ "concept/reciprocal", "concept/variable", "method/substitution" ] }, { "id": "dickson-theory-of-equations-1922/eq-48711b4d4d", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "\\left(y^3 + \\frac{1}{y^3}\\right) + \\left(y^2 + \\frac{1}{y^2}\\right) + \\left(y + \\frac{1}{y }\\right) + 1=0", "name": null, "statement": "Dividing the equation in y by y^3 gives this form, whose brackets are the powers y^k + 1/y^k.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown, a seventh root of unity other than 1" } ], "sympy": "Eq(y**3 + 1/y**3 + y**2 + 1/y**2 + y + 1/y + 1, 0)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/reciprocal", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-d4c872d09a", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "y^2 + \\frac{1}{y^2} = x^2 - 2", "name": null, "statement": "Squaring y + 1/y = x gives y^2 + 1/y^2 in terms of x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "y + 1/y" } ], "sympy": "Eq(y**2 + 1/y**2, x**2 - 2)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/reciprocal", "method/substitution" ] }, { "id": "dickson-theory-of-equations-1922/eq-a64f147991", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "36", "location": "Constructions with Ruler and Compasses", "latex": "y^3 + \\frac{1}{y^3} = x^3 - 3x", "name": null, "statement": "Cubing y + 1/y = x gives y^3 + 1/y^3 in terms of x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "y + 1/y" } ], "sympy": "Eq(y**3 + 1/y**3, x**3 - 3*x)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/reciprocal", "method/substitution" ] }, { "id": "dickson-theory-of-equations-1922/eq-afbce9cf36", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "x_1 = R + \\frac{1}{R} = R + R^6", "name": null, "statement": "The first of the three sums of pairs of seventh roots of unity equals 2 cos(2 pi / 7).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "sum of R and its reciprocal" }, { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(x1, R + 1/R)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/reciprocal", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-18ac9e662d", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "x_2 = R^2 + \\frac{1}{R^2} = R^2 + R^5", "name": null, "statement": "The second of the three sums of pairs of seventh roots of unity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_2", "meaning": "sum of R^2 and its reciprocal" }, { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(x2, R**2 + 1/R**2)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-7556512d7c", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "x_3 = R^3 + \\frac{1}{R^3} = R^3 + R^4", "name": null, "statement": "The third of the three sums of pairs of seventh roots of unity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_3", "meaning": "sum of R^3 and its reciprocal" }, { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(x3, R**3 + 1/R**3)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-afd1a67805", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "x_1 + x_2 + x_3 = R + R^2 + \\dotsb + R^6 = -1", "name": null, "statement": "The sum of the three values x_1, x_2, x_3 equals -1, since R, ..., R^6 are the roots of the equation (13).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1, x_2, x_3", "meaning": "the three sums of pairs of roots" }, { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(x1 + x2 + x3, -1)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-69b49ea4e1", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "x_1 x_2 + x_1 x_3 + x_2 x_3 = 2(R + R^2 + \\dotsb +R^6) = -2", "name": null, "statement": "The sum of pairwise products of x_1, x_2, x_3 equals -2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1, x_2, x_3", "meaning": "the three sums of pairs of roots" } ], "sympy": "Eq(x1*x2 + x1*x3 + x2*x3, -2)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/product", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-56eefb0d98", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "x_1 x_2 x_3 = 2 + R + R^2 + \\dotsb + R^6 = 1", "name": null, "statement": "The product x_1 x_2 x_3 equals 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1, x_2, x_3", "meaning": "the three sums of pairs of roots" } ], "sympy": "Eq(x1*x2*x3, 1)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/product", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-0b11f3e3fa", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "y^n f\\left(\\frac{1}{y}\\right) \\equiv ±f(y)", "name": null, "statement": "For a reciprocal equation f(y)=0 of degree n with constant term c, the reversed polynomial equals plus or minus f(y).", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(y)", "meaning": "a polynomial of degree n with constant term c" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/identity", "concept/polynomial", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-2691c26fd1", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "37", "location": "Constructions with Ruler and Compasses", "latex": "c^2 = 1", "name": null, "statement": "Equating constant terms in the reversed polynomial gives c^2 = 1, so c = ±1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "constant term of the polynomial f(y)" } ], "sympy": "Eq(c**2, 1)", "physics": false, "states": [], "concepts": [ "concept/constant-term", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-dd15919ba8", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "38", "location": "Constructions with Ruler and Compasses", "latex": "f(y) \\equiv y^n ± 1 + p_1(y^{n-1} ± y) + p_2 (y^{n-2} ± y^2) + \\dotsb", "name": null, "statement": "Form of a reciprocal polynomial: the coefficients of terms equidistant from the ends are equal up to sign.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p_i", "meaning": "coefficient of a term in f(y)" }, { "unit": null, "symbol": "n", "meaning": "degree of f" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-036d6e2de2", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "38", "location": "Constructions with Ruler and Compasses", "latex": "y^{n-1} Q \\left(\\frac{1}{y}\\right) \\equiv Q(y)", "name": null, "statement": "The quotient Q(y) = f(y)/(y ± 1) is again reciprocal, with reversal degree n - 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q(y)", "meaning": "quotient f(y)/(y ± 1)" }, { "unit": null, "symbol": "n", "meaning": "degree of f" } ], "sympy": "Eq(y**(n-1)*Q(1/y), Q(y))", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/quotient", "concept/reciprocal" ] }, { "id": "dickson-theory-of-equations-1922/eq-a8012a68aa", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "38", "location": "Constructions with Ruler and Compasses", "latex": "y^{2t} + 1 + c_1 (y^{2t-1} + y) + c_2 (y^{2t-2} + y^2) + \\dotsb + c_{t-1} (y^{t+1} + y^{t-1}) + c_t y^t = 0", "name": null, "statement": "The standard form to which any reciprocal equation of even degree 2t can be reduced.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown" }, { "unit": null, "symbol": "t", "meaning": "half the even degree" }, { "unit": null, "symbol": "c_i", "meaning": "coefficients" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/equation", "concept/reciprocal", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-776c7f347f", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "38", "location": "Constructions with Ruler and Compasses", "latex": "n = 2t+1", "name": null, "statement": "Writing an odd degree n as 2t + 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "degree of the reciprocal equation" }, { "unit": null, "symbol": "t", "meaning": "integer half-degree" } ], "sympy": "Eq(n, 2*t + 1)", "physics": false, "states": [], "concepts": [ "concept/degree", "concept/integer" ] }, { "id": "dickson-theory-of-equations-1922/eq-79adf0e5ff", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "38", "location": "Constructions with Ruler and Compasses", "latex": "y^k + \\frac{1}{y^k} = x \\left(y^{k-1} + \\frac{1}{y^{k-1}}\\right) - \\left(y^{k-2} + \\frac{1}{y^{k-2}}\\right)", "name": null, "statement": "Recurrence giving y^k + 1/y^k in terms of x = y + 1/y and the two previous powers.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "y + 1/y" }, { "unit": null, "symbol": "k", "meaning": "positive integer power" } ], "sympy": "Eq(y**k + 1/y**k, x*(y**(k-1) + 1/y**(k-1)) - (y**(k-2) + 1/y**(k-2)))", "physics": false, "states": [], "concepts": [ "concept/power", "concept/reciprocal", "method/substitution" ] }, { "id": "dickson-theory-of-equations-1922/eq-bf244731b0", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Constructions with Ruler and Compasses", "latex": "x (x^3-3x) - (x^2-2) = x^4 - 4x^2 + 2", "name": null, "statement": "Applying the recurrence gives y^4 + 1/y^4 = x^4 - 4x^2 + 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "y + 1/y" } ], "sympy": "Eq(x*(x**3 - 3*x) - (x**2 - 2), x**4 - 4*x**2 + 2)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/reciprocal", "method/substitution" ] }, { "id": "dickson-theory-of-equations-1922/eq-4d3eaab8e1", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Constructions with Ruler and Compasses", "latex": "R = \\cos\\frac{ 2\\pi}{9} + i \\sin\\frac{ 2\\pi}{9}", "name": null, "statement": "R is the complex ninth root of unity with argument 2 pi / 9.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "a primitive ninth root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(R, cos(2*pi/9) + I*sin(2*pi/9))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/regular-polygon", "concept/root-of-an-equation", "concept/trigonometric-form-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-9b36a1ffcb", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Constructions with Ruler and Compasses", "latex": "\\frac{y^9 - 1}{y^3 - 1} = y^6 + y^3 + 1 = 0", "name": null, "statement": "The primitive ninth roots of unity are roots of y^6 + y^3 + 1 = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown, a primitive ninth root of unity" } ], "sympy": "Eq(y**6 + y**3 + 1, 0)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-d27c7b37fb", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "40", "location": "Constructions with Ruler and Compasses", "latex": "R^8 = R", "name": null, "statement": "Since R^7 = 1 for the seventh root of unity, the fourth power of the ordering g = 2 returns to R, so g = 2 is rejected.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(R**8, R)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-739a438991", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z_1 = R + R^2 + R^4", "name": null, "statement": "Period of three terms formed from alternate terms of the ordering R, R^3, R^2, R^6, R^4, R^5.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z_1", "meaning": "period of three seventh roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(z1, R + R**2 + R**4)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-147b3b4862", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z_2 = R^3 + R^6 + R^5", "name": null, "statement": "The second period of three terms of the seventh roots of unity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z_2", "meaning": "period of three seventh roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(z2, R**3 + R**6 + R**5)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-c151e5de6f", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z_1 z_2 = 3 + R + \\dotsb + R^6 = 2", "name": null, "statement": "The product of the two periods of three terms equals 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z_1, z_2", "meaning": "the two periods of three terms" }, { "unit": null, "symbol": "R", "meaning": "seventh root of unity" } ], "sympy": "Eq(z1*z2, 2)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-3db1099072", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z^2 + z + 2 = 0", "name": null, "statement": "The periods z_1 and z_2 are the roots of this quadratic (their sum is -1 and product 2).", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown, a period of seventh roots of unity" } ], "sympy": "Eq(z**2 + z + 2, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-a5b3b0f054", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "w^3 - z_1w^2 + z_2w - 1 = 0", "name": null, "statement": "The numbers R, R^2, R^4 are the roots of this cubic whose coefficients are the periods z_1 and z_2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "w", "meaning": "unknown, one of R, R^2, R^4" }, { "unit": null, "symbol": "z_1, z_2", "meaning": "periods of seventh roots of unity" } ], "sympy": "Eq(w**3 - z1*w**2 + z2*w - 1, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/cubic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-2fb1f035ce", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "\\frac{R^{17} - 1}{R - 1} = R^{16} + R^{15} + \\dotsb + R + 1 = 0", "name": null, "statement": "The seventeenth roots of unity other than 1 are roots of this reciprocal polynomial.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "a root of x^17 = 1 other than 1" } ], "sympy": "Eq((R**17 - 1)/(R - 1), 0)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-an-equation", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-11b5480f09", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "y_1 + y_2 = -1", "name": null, "statement": "The two periods of eight terms of the 17th roots of unity sum to -1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1, y_2", "meaning": "periods of eight seventeenth roots of unity" } ], "sympy": "Eq(y1 + y2, -1)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-31bfd5a67c", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "y_1 y_2 = 4(R + \\dotsb + R^{16}) = -4", "name": null, "statement": "The product of the two eight-term periods equals -4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1, y_2", "meaning": "periods of eight seventeenth roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": "Eq(y1*y2, -4)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-b7e47d7070", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "y^2 + y - 4 = 0", "name": null, "statement": "The periods y_1 and y_2 of the seventeenth roots of unity are roots of this quadratic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown, a period of seventeenth roots of unity" } ], "sympy": "Eq(y**2 + y - 4, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-a46a7513f5", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "R + R^9 + R^{13} + R^{15} + R^{16} + R^8 + R^4 + R^2", "name": null, "statement": "The period y_1 of eight seventeenth roots of unity (the sum R + ... + R^2 taken over the even-position terms of the cube-ordering).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-aa219b353a", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "R^3 + R^{10} + R^5 + R^{11} + R^{14} + R^7 + R^{12} + R^6", "name": null, "statement": "The period y_2 of eight seventeenth roots of unity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-ca6a3c5cbf", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z_1 = R + R^{13} + R^{16} + R^4", "name": null, "statement": "Period of four seventeenth roots of unity, obtained from alternate terms of y_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z_1", "meaning": "period of four seventeenth roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": "Eq(z1, R + R**13 + R**16 + R**4)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-a09c5c1387", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z_2 = R^9 + R^{15} + R^8 + R^2", "name": null, "statement": "Period of four seventeenth roots of unity, the other half of y_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z_2", "meaning": "period of four seventeenth roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": "Eq(z2, R**9 + R**15 + R**8 + R**2)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-f3ffcaf514", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "w_1 = R^3 + R^5 + R^{14} + R^{12}", "name": null, "statement": "Period of four seventeenth roots of unity, obtained from alternate terms of y_2.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "w_1", "meaning": "period of four seventeenth roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": "Eq(w1, R**3 + R**5 + R**14 + R**12)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-412c876665", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "w_2 = R^{10} + R^{11} + R^7 + R^6", "name": null, "statement": "Period of four seventeenth roots of unity, the other half of y_2.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "w_2", "meaning": "period of four seventeenth roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": "Eq(w2, R**10 + R**11 + R**7 + R**6)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-17846b7dfe", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z_1 + z_2 = y_1", "name": null, "statement": "The two four-term periods z_1, z_2 add to the eight-term period y_1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z_1, z_2", "meaning": "periods of four seventeenth roots of unity" }, { "unit": null, "symbol": "y_1", "meaning": "period of eight seventeenth roots of unity" } ], "sympy": "Eq(z1 + z2, y1)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-a1dcae5abc", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z_1 z_2 = w_1 w_2 = -1", "name": null, "statement": "The products of the paired four-term periods both equal -1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z_1, z_2, w_1, w_2", "meaning": "periods of four seventeenth roots of unity" } ], "sympy": "Eq(z1*z2, -1)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-483c5e4f1c", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "z^2 - y_1 z - 1 = 0", "name": null, "statement": "The periods z_1 and z_2 are the roots of this quadratic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown, a four-term period" }, { "unit": null, "symbol": "y_1", "meaning": "eight-term period" } ], "sympy": "Eq(z**2 - y1*z - 1, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-dddb132073", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "41", "location": "Constructions with Ruler and Compasses", "latex": "w^2 - y_2 w - 1 = 0", "name": null, "statement": "The periods w_1 and w_2 are the roots of this quadratic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "w", "meaning": "unknown, a four-term period" }, { "unit": null, "symbol": "y_2", "meaning": "eight-term period" } ], "sympy": "Eq(w**2 - y2*w - 1, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-dfc34646c5", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "v_1 = R + R^{16}", "name": null, "statement": "Period of two seventeenth roots of unity, from alternate terms of z_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "v_1", "meaning": "period of two seventeenth roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": "Eq(v1, R + R**16)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-d470d87629", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "v_2 = R^{13} + R^4", "name": null, "statement": "Second period of two seventeenth roots of unity, from alternate terms of z_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "v_2", "meaning": "period of two seventeenth roots of unity" }, { "unit": null, "symbol": "R", "meaning": "seventeenth root of unity" } ], "sympy": "Eq(v2, R**13 + R**4)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-de0ee14927", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "v_1 + v_2 = z_1", "name": null, "statement": "The two periods v_1, v_2 add to z_1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v_1, v_2", "meaning": "periods of two seventeenth roots of unity" }, { "unit": null, "symbol": "z_1", "meaning": "four-term period" } ], "sympy": "Eq(v1 + v2, z1)", "physics": false, "states": [], "concepts": [ "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-28d7fcac2f", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "v_1v_2 = w_1", "name": null, "statement": "The product of v_1 and v_2 equals the four-term period w_1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v_1, v_2", "meaning": "periods of two seventeenth roots of unity" }, { "unit": null, "symbol": "w_1", "meaning": "four-term period" } ], "sympy": "Eq(v1*v2, w1)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-87a089ce8d", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "v^2 - z_1v + w_1 = 0", "name": null, "statement": "The periods v_1 and v_2 are the roots of this quadratic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "unknown, a two-term period" }, { "unit": null, "symbol": "z_1, w_1", "meaning": "four-term periods" } ], "sympy": "Eq(v**2 - z1*v + w1, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-5f1212c0c5", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "\\rho^2 - v_1\\rho + 1 = 0", "name": null, "statement": "R and R^16 are the roots of this quadratic, so R can be found by solving quadratics.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ρ", "meaning": "unknown, a seventeenth root of unity" }, { "unit": null, "symbol": "v_1", "meaning": "two-term period" } ], "sympy": "Eq(rho**2 - v1*rho + 1, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-57a1d6f8a9", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "43", "location": "Constructions with Ruler and Compasses", "latex": "v_1 = 2 \\cos\\frac{2\\pi}{17}", "name": null, "statement": "The larger period v_1 equals twice the cosine of 2 pi / 17, so the angle 2 pi / 17 is constructible.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v_1", "meaning": "larger two-term period" } ], "sympy": "Eq(v1, 2*cos(2*pi/17))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/regular-polygon", "concept/root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-31020f0b49", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "2 \\cos \\frac{6\\pi}{17} + 2 \\cos \\frac{10\\pi}{17} = 2 \\cos \\frac{6\\pi}{17} - 2 \\cos \\frac{7\\pi}{17}", "name": null, "statement": "Since cos(10 pi/17) = -cos(7 pi/17), the sum of the two cosines equals the expression with -2 cos(7 pi/17), which shows w_1 > 0.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "π", "meaning": "the circle constant" } ], "sympy": "Eq(2*cos(6*pi/17) + 2*cos(10*pi/17), 2*cos(6*pi/17) - 2*cos(7*pi/17))", "physics": false, "states": [], "concepts": [ "concept/cosine", "quantity/angle", "theorem/trigonometric-identity" ] }, { "id": "dickson-theory-of-equations-1922/eq-2e0afa26fe", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "42", "location": "Constructions with Ruler and Compasses", "latex": "2 \\cos \\frac{6\\pi}{17} + 2 \\cos \\frac{10\\pi}{17} + 2 \\cos \\frac{12\\pi}{17} + 2 \\cos \\frac{14\\pi}{17} < 0", "name": null, "statement": "The period y_2 is negative, since only the first cosine is positive and it is numerically less than the third.", "kind": "result", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/root-of-unity", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-9236ec48cf", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "43", "location": "Constructions with Ruler and Compasses", "latex": "OE = \\tfrac{1}{4} \\sqrt{17}", "name": null, "statement": "Length OE, the radius of the circle through which the point of AS is located for the 17-gon construction; equals sqrt(17)/4 for unit radius.", "kind": "result", "symbols": [ { "unit": "unit length", "symbol": "OE", "meaning": "distance from centre O to point E on the tangent AS" } ], "sympy": "Eq(OE, sqrt(17)/4)", "physics": false, "states": [], "concepts": [ "concept/construction", "concept/radius-of-a-regular-polygon", "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-55a3a70e08", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Constructions with Ruler and Compasses", "latex": "\\cos LOP = OL = \\cos\\frac{2\\pi}{17}", "name": null, "statement": "The perpendicular bisector of OM meets the unit circle at P so that angle LOP equals 2 pi / 17, which gives the side of the 17-gon.", "kind": "result", "symbols": [ { "unit": "degree of angle", "symbol": "LOP", "meaning": "angle at O between OL and OP" } ], "sympy": "Eq(cos(LOP), cos(2*pi/17))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/perpendicular-bisector", "concept/regular-polygon", "quantity/angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-c2b1e0e29b", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "32", "location": "Constructions with Ruler and Compasses", "latex": "\\tfrac{1}{2}\\sqrt{10 - 2\\sqrt{5}}", "name": null, "statement": "placeholder not used", "kind": "result", "symbols": [], "sympy": null, "physics": false, "states": [], "concepts": [] }, { "id": "dickson-theory-of-equations-1922/eq-9d40885121", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "33", "location": "Constructions with Ruler and Compasses", "latex": "st = \\sqrt{5}", "name": null, "statement": "The product of s (side of the pentagon) and t equals sqrt(5).", "kind": "result", "symbols": [ { "unit": "unit length", "symbol": "s", "meaning": "side of a regular pentagon inscribed in a unit circle" }, { "unit": null, "symbol": "t", "meaning": "half-root (1/2) sqrt(10 + 2 sqrt 5)" } ], "sympy": "Eq(s*t, sqrt(5))", "physics": false, "states": [], "concepts": [ "concept/pentagon", "concept/product", "concept/radical-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-812d01f5d5", "chapter": "dickson-theory-of-equations-1922/ch-iii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "33", "location": "Constructions with Ruler and Compasses", "latex": "t = \\tfrac{1}{2} \\sqrt{10 + 2\\sqrt{5}}", "name": null, "statement": "Definition of t as half the square root of 10 + 2 sqrt(5), a number of order 2.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "t", "meaning": "half of sqrt(10 + 2 sqrt 5)" } ], "sympy": "Eq(t, sqrt(10 + 2*sqrt(5))/2)", "physics": false, "states": [], "concepts": [ "concept/pentagon", "concept/radical-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-e7621149f0", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x^3 + bx^2 + cx + d = 0", "name": null, "statement": "The general cubic equation, with the coefficient of x^3 equal to one.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x" }, { "unit": null, "symbol": "d", "meaning": "constant term" } ], "sympy": "Eq(x**3 + b*x**2 + c*x + d, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/cubic-equation", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-3c8c1052e2", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y^3 + py + q = 0", "name": null, "statement": "The reduced cubic equation, which lacks the square of the unknown.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the unknown of the reduced cubic" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" } ], "sympy": "Eq(y**3 + p*y + q, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/cubic-equation", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-6596361c2f", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "p = c - \\frac{b^2}{3}", "name": null, "statement": "The coefficient p of the reduced cubic is c minus one third of b squared.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^2 in the general cubic" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x in the general cubic" } ], "sympy": "Eq(p, c - b**2/3)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-4e83469fb5", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "q = d - \\frac{bc}{3} + \\frac{2b^3}{27}", "name": null, "statement": "The constant q of the reduced cubic is expressed through the coefficients of the general cubic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^2 in the general cubic" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x in the general cubic" }, { "unit": null, "symbol": "d", "meaning": "constant term of the general cubic" } ], "sympy": "Eq(q, d - b*c/3 + 2*b**3/27)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-98cba7d65d", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_1 = y_1 - \\frac{b}{3}", "name": null, "statement": "Each root of the general cubic is the corresponding root of the reduced cubic shifted by minus b/3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a root of the general cubic" }, { "unit": null, "symbol": "y_1", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^2 in the general cubic" } ], "sympy": "Eq(x_1, y_1 - b/3)", "physics": false, "states": [], "concepts": [ "concept/reduced-cubic-equation", "concept/root-of-an-equation", "method/transposing-the-terms" ] }, { "id": "dickson-theory-of-equations-1922/eq-7b5584235b", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y = z - \\frac{p}{3z}", "name": null, "statement": "Vieta's substitution that turns the reduced cubic into an equation in z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown of the reduced cubic" }, { "unit": null, "symbol": "z", "meaning": "auxiliary unknown of the substitution" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" } ], "sympy": "Eq(y, z - p/(3*z))", "physics": false, "states": [], "concepts": [ "concept/reduced-cubic-equation", "method/substitution" ] }, { "id": "dickson-theory-of-equations-1922/eq-09f265c6d4", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "z^6 + qz^3 - \\frac{p^3}{27} = 0", "name": null, "statement": "Substituting the auxiliary form into the reduced cubic gives a sextic that is quadratic in z^3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "auxiliary unknown" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" } ], "sympy": "Eq(z**6 + q*z**3 - p**3/27, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-6925d38e51", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "45", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "R = \\left(\\frac{p}{3}\\right)^3 + \\left(\\frac{q}{2}\\right)^2", "name": null, "statement": "R is defined as the sum of the cube of p/3 and the square of q/2; its sign decides the number of real roots.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "discriminant-like quantity of the reduced cubic" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" } ], "sympy": "Eq(R, (p/3)**3 + (q/2)**2)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-99077da007", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\omega = -\\tfrac{1}{2} + \\tfrac{1}{2} \\sqrt{3}i", "name": null, "statement": "omega is one of the imaginary cube roots of unity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(omega, Rational(-1,2) + sqrt(3)*I/2)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cube-root-of-unity", "concept/imaginary-root" ] }, { "id": "dickson-theory-of-equations-1922/eq-2de7d35a68", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\omega^2 = -\\tfrac{1}{2} - \\tfrac{1}{2} \\sqrt{3}i", "name": null, "statement": "omega squared is the other imaginary cube root of unity, the conjugate of omega.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(omega**2, Rational(-1,2) - sqrt(3)*I/2)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/conjugate-complex-numbers", "concept/cube-root-of-unity" ] }, { "id": "dickson-theory-of-equations-1922/eq-838f091d7a", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "A = \\sqrt[3]{-\\frac{q}{2} + \\sqrt{R}}", "name": null, "statement": "A is a chosen cube root of minus q/2 plus the square root of R.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A", "meaning": "chosen cube root in Cardan's formulas" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" }, { "unit": null, "symbol": "R", "meaning": "quantity (p/3)^3 + (q/2)^2" } ], "sympy": "Eq(A, (-q/2 + sqrt(R))**Rational(1,3))", "physics": false, "states": [], "concepts": [ "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-5e33576228", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "B = \\sqrt[3]{-\\frac{q}{2} - \\sqrt{R}}", "name": null, "statement": "B is a chosen cube root of minus q/2 minus the square root of R.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "B", "meaning": "chosen cube root in Cardan's formulas" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" }, { "unit": null, "symbol": "R", "meaning": "quantity (p/3)^3 + (q/2)^2" } ], "sympy": "Eq(B, (-q/2 - sqrt(R))**Rational(1,3))", "physics": false, "states": [], "concepts": [ "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-137814bd93", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "AB = -\\frac{p}{3}", "name": null, "statement": "The chosen cube roots A and B have product minus p/3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "chosen cube root" }, { "unit": null, "symbol": "B", "meaning": "chosen cube root" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" } ], "sympy": "Eq(A*B, -p/3)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-a2d7477e1e", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_1 = A + B", "name": "Cardan's formulas", "statement": "The first root of the reduced cubic is the sum A + B.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "A", "meaning": "chosen cube root" }, { "unit": null, "symbol": "B", "meaning": "chosen cube root" } ], "sympy": "Eq(y_1, A + B)", "physics": false, "states": [ "theorem/cardan-s-formulas" ], "concepts": [ "concept/root-of-an-equation", "concept/sum-of-the-roots" ] }, { "id": "dickson-theory-of-equations-1922/eq-e057477cfa", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_2 = \\omega A + \\omega^2 B", "name": "Cardan's formulas", "statement": "The second root of the reduced cubic is omega times A plus omega squared times B.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y_2", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" }, { "unit": null, "symbol": "A", "meaning": "chosen cube root" }, { "unit": null, "symbol": "B", "meaning": "chosen cube root" } ], "sympy": "Eq(y_2, omega*A + omega**2*B)", "physics": false, "states": [ "theorem/cardan-s-formulas" ], "concepts": [ "concept/cube-root-of-unity", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-2cb1eaaa80", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_3 = \\omega^2 A + \\omega B", "name": "Cardan's formulas", "statement": "The third root of the reduced cubic is omega squared times A plus omega times B.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y_3", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" }, { "unit": null, "symbol": "A", "meaning": "chosen cube root" }, { "unit": null, "symbol": "B", "meaning": "chosen cube root" } ], "sympy": "Eq(y_3, omega**2*A + omega*B)", "physics": false, "states": [ "theorem/cardan-s-formulas" ], "concepts": [ "concept/cube-root-of-unity", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-129f5917fd", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(y_1 - y_2)^2 (y_1 - y_3)^2 (y_2 - y_3)^2 = -4p^3 - 27q^2", "name": null, "statement": "The product of the squares of the differences of the roots of the reduced cubic equals minus four p cubed minus 27 q squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "y_2", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "y_3", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" } ], "sympy": "Eq((y_1-y_2)**2*(y_1-y_3)**2*(y_2-y_3)**2, -4*p**3 - 27*q**2)", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/discriminant", "concept/reduced-cubic-equation", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-8583a9fea2", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(x-1)(x-\\omega)(x-\\omega^2) \\equiv x^3 - 1", "name": null, "statement": "The cube roots of unity 1, omega, omega squared are the roots of x^3 - 1, identically in x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" } ], "sympy": "Eq((x-1)*(x-omega)*(x-omega**2), x**3 - 1)", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/identity" ] }, { "id": "dickson-theory-of-equations-1922/eq-5fc07deaa2", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(A-B)(A-\\omega B)(A-\\omega^2 B) = A^3 - B^3 = 2 \\sqrt{R}", "name": null, "statement": "The product (A-B)(A-omega B)(A-omega^2 B) equals A cubed minus B cubed, which equals 2 times the square root of R.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "chosen cube root" }, { "unit": null, "symbol": "B", "meaning": "chosen cube root" }, { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" }, { "unit": null, "symbol": "R", "meaning": "quantity (p/3)^3 + (q/2)^2" } ], "sympy": "Eq((A-B)*(A-omega*B)*(A-omega**2*B), 2*sqrt(R))", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/product", "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-9d97af5edc", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(1-\\omega)(1-\\omega^2) = 3", "name": null, "statement": "The product of one minus omega and one minus omega squared equals 3.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" } ], "sympy": "Eq((1-omega)*(1-omega**2), 3)", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/product" ] }, { "id": "dickson-theory-of-equations-1922/eq-77315f5426", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\omega - \\omega^2 = \\sqrt{3}i", "name": null, "statement": "The difference of the two imaginary cube roots of unity is root 3 times i.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "omega", "meaning": "imaginary cube root of unity" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(omega - omega**2, sqrt(3)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cube-root-of-unity", "concept/difference" ] }, { "id": "dickson-theory-of-equations-1922/eq-74c69d1aeb", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(y_1-y_2)(y_1-y_3)(y_2-y_3) = 6\\sqrt{3}\\sqrt{R}i", "name": null, "statement": "The product of the differences of the three roots of the reduced cubic equals 6 root 3 times root R times i.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "y_2", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "y_3", "meaning": "a root of the reduced cubic" }, { "unit": null, "symbol": "R", "meaning": "quantity (p/3)^3 + (q/2)^2" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((y_1-y_2)*(y_1-y_3)*(y_2-y_3), 6*sqrt(3)*sqrt(R)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/difference", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-6471c37b12", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "-108R = -4p^3 - 27q^2", "name": null, "statement": "The discriminant of the reduced cubic equals -108 R.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "quantity (p/3)^3 + (q/2)^2" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" } ], "sympy": "Eq(-108*R, -4*p**3 - 27*q**2)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-f0881680b8", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\Delta = 18bcd - 4b^3 d + b^2 c^2 - 4c^3 - 27d^2", "name": null, "statement": "The discriminant of the general cubic is expressed directly in its coefficients b, c, d.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "discriminant of the general cubic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x" }, { "unit": null, "symbol": "d", "meaning": "constant term" } ], "sympy": "Eq(Delta, 18*b*c*d - 4*b**3*d + b**2*c**2 - 4*c**3 - 27*d**2)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/cubic-equation", "concept/discriminant" ] }, { "id": "dickson-theory-of-equations-1922/eq-8034c0ec51", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "ax^3 + bx^2 + cx +d = 0 \\quad (a \\neq 0)", "name": null, "statement": "A cubic equation whose leading coefficient a is not required to be one.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^3, not zero" }, { "unit": null, "symbol": "x", "meaning": "the unknown" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x" }, { "unit": null, "symbol": "d", "meaning": "constant term" } ], "sympy": "Eq(a*x**3 + b*x**2 + c*x + d, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-db2a962045", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "47", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "a^4 P = 18 abcd - 4b^3 d + b^2 c^2 - 4ac^3 - 27a^2 d^2", "name": null, "statement": "For a cubic with leading coefficient a, a to the fourth times the product P of squared root differences equals this expression in a, b, c, d; this expression is the discriminant of that cubic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "product of squares of differences of the roots" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^3" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x" }, { "unit": null, "symbol": "d", "meaning": "constant term" } ], "sympy": "Eq(a**4*P, 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/discriminant", "concept/product" ] }, { "id": "dickson-theory-of-equations-1922/eq-6a497c2f9d", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "50", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x^4 +bx^3 +cx^2 +dx+e=0", "name": "quartic equation", "statement": "The general quartic equation, with leading coefficient one.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "d", "meaning": "coefficient of x" }, { "unit": null, "symbol": "e", "meaning": "constant term" } ], "sympy": "Eq(x**4 + b*x**3 + c*x**2 + d*x + e, 0)", "physics": false, "states": [ "concept/quartic-equation" ], "concepts": [ "concept/coefficient", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-91be38c8e2", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "50", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(x^2 + \\tfrac{1}{2}bx + \\tfrac{1}{2}y)^2 = (\\tfrac{1}{4}b^2 - c + y)x^2 + (\\tfrac{1}{2}by - d)x + \\tfrac{1}{4}y^2 - e", "name": null, "statement": "Adding the same terms to both sides of the quartic makes the left side a perfect square, leaving a quadratic in x on the right, for any y.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown of the quartic" }, { "unit": null, "symbol": "y", "meaning": "auxiliary parameter chosen by Ferrari's method" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "d", "meaning": "coefficient of x" }, { "unit": null, "symbol": "e", "meaning": "constant term" } ], "sympy": "Eq((x**2 + b*x/2 + y/2)**2, (b**2/4 - c + y)*x**2 + (b*y/2 - d)*x + y**2/4 - e)", "physics": false, "states": [], "concepts": [ "concept/perfect-square", "concept/quadratic-equation", "method/completing-the-square", "method/ferrari-s-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-265ee3ae11", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "50", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(\\tfrac{1}{2}by - d)^2 - 4(\\tfrac{1}{4}b^2 - c + y)(\\tfrac{1}{4}y^2 - e) = 0", "name": null, "statement": "The right side of the previous identity is a perfect square exactly when its discriminant is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "auxiliary parameter of Ferrari's method" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "d", "meaning": "coefficient of x" }, { "unit": null, "symbol": "e", "meaning": "constant term" } ], "sympy": "Eq((b*y/2 - d)**2 - 4*(b**2/4 - c + y)*(y**2/4 - e), 0)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/perfect-square", "method/ferrari-s-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-5aaac54a36", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "50", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y^3 - cy^2 + (bd - 4e)y - b^{2}e + 4ce - d^2 = 0", "name": "resolvent cubic equation", "statement": "The resolvent cubic of the quartic, whose roots y allow the quartic to be factored into quadratics.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown of the resolvent cubic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3 in the quartic" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2 in the quartic" }, { "unit": null, "symbol": "d", "meaning": "coefficient of x in the quartic" }, { "unit": null, "symbol": "e", "meaning": "constant term of the quartic" } ], "sympy": "Eq(y**3 - c*y**2 + (b*d - 4*e)*y - b**2*e + 4*c*e - d**2, 0)", "physics": false, "states": [ "concept/resolvent-cubic" ], "concepts": [ "concept/cubic-equation", "concept/quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-00e50a8546", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "50", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x^2 + \\tfrac{1}{2}bx + \\tfrac{1}{2}y = mx+n", "name": null, "statement": "One of the two quadratic equations obtained by taking the square root of the perfect square, with m and n the coefficients of the linear function.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown of the quartic" }, { "unit": null, "symbol": "y", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3" }, { "unit": null, "symbol": "m", "meaning": "coefficient of x in the linear function" }, { "unit": null, "symbol": "n", "meaning": "constant term of the linear function" } ], "sympy": "Eq(x**2 + b*x/2 + y/2, m*x + n)", "physics": false, "states": [], "concepts": [ "concept/linear-function", "concept/quadratic-equation", "method/ferrari-s-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-7764c3a6b3", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_1 = x_1 x_2 + x_3 x_4", "name": null, "statement": "The first root of the resolvent cubic is the sum of the products of the roots of the quartic paired as (x1,x2) and (x3,x4).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" } ], "sympy": "Eq(y_1, x_1*x_2 + x_3*x_4)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/resolvent-cubic", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-a11aa4e10c", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_2 = x_1 x_3 + x_2 x_4", "name": null, "statement": "The second root of the resolvent cubic pairs the roots as (x1,x3) and (x2,x4).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y_2", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" } ], "sympy": "Eq(y_2, x_1*x_3 + x_2*x_4)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/resolvent-cubic", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-93454282cb", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_3 = x_1 x_4 + x_2 x_3", "name": null, "statement": "The third root of the resolvent cubic pairs the roots as (x1,x4) and (x2,x3).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y_3", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" } ], "sympy": "Eq(y_3, x_1*x_4 + x_2*x_3)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/resolvent-cubic", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-1cfa74d7d1", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_1 x_2 = \\tfrac{1}{2} y_1 - n", "name": null, "statement": "The product of the first pair of roots of the quartic equals half of y1 minus n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "y_1", "meaning": "root of the resolvent cubic used in Ferrari's method" }, { "unit": null, "symbol": "n", "meaning": "constant term of the linear function" } ], "sympy": "Eq(x_1*x_2, y_1/2 - n)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/resolvent-cubic", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-44bd77719d", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_3 x_4 = \\tfrac{1}{2} y_1 + n", "name": null, "statement": "The product of the second pair of roots of the quartic equals half of y1 plus n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "y_1", "meaning": "root of the resolvent cubic used in Ferrari's method" }, { "unit": null, "symbol": "n", "meaning": "constant term of the linear function" } ], "sympy": "Eq(x_3*x_4, y_1/2 + n)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/resolvent-cubic", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-4e6014cb4b", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_1 x_2 + x_3 x_4 = y_1", "name": null, "statement": "The sum of the two pair-products of the quartic roots equals y1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "y_1", "meaning": "root of the resolvent cubic" } ], "sympy": "Eq(x_1*x_2 + x_3*x_4, y_1)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/resolvent-cubic", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-736c0060d8", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_1 + x_2 + x_3 + x_4 = -b", "name": null, "statement": "The sum of the four roots of the quartic equals minus the coefficient b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3 in the quartic" } ], "sympy": "Eq(x_1 + x_2 + x_3 + x_4, -b)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/root-of-an-equation", "concept/sum-of-the-roots" ] }, { "id": "dickson-theory-of-equations-1922/eq-3ce9cec79a", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4 = -d", "name": null, "statement": "The sum of the triple products of the quartic roots equals minus d.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "d", "meaning": "coefficient of x in the quartic" } ], "sympy": "Eq(x_1*x_2*x_3 + x_1*x_2*x_4 + x_1*x_3*x_4 + x_2*x_3*x_4, -d)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/product", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-ee84fe1e31", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4 = c", "name": null, "statement": "The sum of the pairwise products of the quartic roots equals c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2 in the quartic" } ], "sympy": "Eq(x_1*x_2 + x_1*x_3 + x_1*x_4 + x_2*x_3 + x_2*x_4 + x_3*x_4, c)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/product", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "dickson-theory-of-equations-1922/eq-7b4fcb73f0", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "x_1 x_2 x_3 x_4 = e", "name": null, "statement": "The product of the four roots of the quartic equals e.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "e", "meaning": "constant term of the quartic" } ], "sympy": "Eq(x_1*x_2*x_3*x_4, e)", "physics": false, "states": [], "concepts": [ "concept/constant-term", "concept/product", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-2174b10c59", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_1 + y_2 + y_3 = c", "name": null, "statement": "The sum of the roots of the resolvent cubic equals c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "y_2", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "y_3", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2 in the quartic" } ], "sympy": "Eq(y_1 + y_2 + y_3, c)", "physics": false, "states": [], "concepts": [ "concept/resolvent-cubic", "concept/root-of-an-equation", "concept/sum-of-the-roots" ] }, { "id": "dickson-theory-of-equations-1922/eq-ad4801ebf6", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\Delta = ( x_1 - x_2 )^2 ( x_1 - x_3 )^2 ( x_1 - x_4 )^2 ( x_2 - x_3 )^2 ( x_2 - x_4 )^2 ( x_3 - x_4 )^2", "name": null, "statement": "The discriminant of the quartic is defined as the product of the squares of the differences of its four roots.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "discriminant of the quartic" }, { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" } ], "sympy": "Eq(Delta, (x_1-x_2)**2*(x_1-x_3)**2*(x_1-x_4)**2*(x_2-x_3)**2*(x_2-x_4)**2*(x_3-x_4)**2)", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/discriminant", "concept/quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-1f55d0cb8b", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "52", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "y_1 - y_2 = (x_1-x_4)(x_2-x_3)", "name": null, "statement": "The difference of two resolvent cubic roots factors as a product of root differences of the quartic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "y_2", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "x_1", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_2", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_3", "meaning": "a root of the quartic" }, { "unit": null, "symbol": "x_4", "meaning": "a root of the quartic" } ], "sympy": "Eq(y_1 - y_2, (x_1-x_4)*(x_2-x_3))", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/resolvent-cubic", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-21c1b15cff", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "52", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(y_1-y_2)^2 (y_1-y_3)^2 (y_2- y_3)^2 = \\Delta", "name": null, "statement": "The discriminant of the quartic equals the discriminant of its resolvent cubic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "discriminant of the quartic" }, { "unit": null, "symbol": "y_1", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "y_2", "meaning": "root of the resolvent cubic" }, { "unit": null, "symbol": "y_3", "meaning": "root of the resolvent cubic" } ], "sympy": "Eq((y_1-y_2)**2*(y_1-y_3)**2*(y_2-y_3)**2, Delta)", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/discriminant", "concept/resolvent-cubic" ] }, { "id": "dickson-theory-of-equations-1922/eq-c7221081c9", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "52", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "p = bd - 4e - \\tfrac{1}{3} c^2", "name": null, "statement": "The coefficient p of the reduced cubic obtained from the resolvent cubic of the quartic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p", "meaning": "coefficient of Y in the reduced cubic of the resolvent" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3 in the quartic" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2 in the quartic" }, { "unit": null, "symbol": "d", "meaning": "coefficient of x in the quartic" }, { "unit": null, "symbol": "e", "meaning": "constant term of the quartic" } ], "sympy": "Eq(p, b*d - 4*e - c**2/3)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/reduced-cubic-equation", "concept/resolvent-cubic" ] }, { "id": "dickson-theory-of-equations-1922/eq-cac9022cf1", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "52", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "q = -b^2 e + \\tfrac{1}{3} bcd + \\tfrac{8}{3} ce - d^2 - \\tfrac{2}{27} c^3", "name": null, "statement": "The constant q of the reduced cubic obtained from the resolvent cubic of the quartic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic of the resolvent" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3 in the quartic" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^2 in the quartic" }, { "unit": null, "symbol": "d", "meaning": "coefficient of x in the quartic" }, { "unit": null, "symbol": "e", "meaning": "constant term of the quartic" } ], "sympy": "Eq(q, -b**2*e + b*c*d/3 + 8*c*e/3 - d**2 - 2*c**3/27)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/reduced-cubic-equation", "concept/resolvent-cubic" ] }, { "id": "dickson-theory-of-equations-1922/eq-439a41945c", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "52", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "z^4 + qz^2 + rz + s = 0", "name": "reduced quartic equation", "statement": "The quartic with its z-cubed term removed, the starting point for Descartes' solution.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown of the reduced quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z" }, { "unit": null, "symbol": "s", "meaning": "constant term" } ], "sympy": "Eq(z**4 + q*z**2 + r*z + s, 0)", "physics": false, "states": [ "concept/reduced-quartic-equation" ], "concepts": [ "concept/coefficient", "concept/quartic-equation", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-91019bd7ed", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "52", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "(z^2 + 2kz + l)(z^2 - 2kz + m) = z^4 + (l + m - 4k^2)z^2 + 2k(m - l)z + lm", "name": null, "statement": "The product of two quadratic factors expands to a quartic with the stated coefficients.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown" }, { "unit": null, "symbol": "k", "meaning": "auxiliary constant of the factorization" }, { "unit": null, "symbol": "l", "meaning": "auxiliary constant of the factorization" }, { "unit": null, "symbol": "m", "meaning": "auxiliary constant of the factorization" } ], "sympy": "Eq((z**2 + 2*k*z + l)*(z**2 - 2*k*z + m), z**4 + (l + m - 4*k**2)*z**2 + 2*k*(m - l)*z + l*m)", "physics": false, "states": [], "concepts": [ "concept/factor", "concept/product", "concept/quadratic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-dab3b0efc5", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "52", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "64k^6 + 32qk^4 + 4(q^2 - 4s)k^2 - r^2 = 0", "name": null, "statement": "The cubic in k squared that yields a factorization of the reduced quartic into quadratics (Descartes' resolvent).", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "auxiliary constant of the factorization" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in the reduced quartic" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in the reduced quartic" }, { "unit": null, "symbol": "s", "meaning": "constant term of the reduced quartic" } ], "sympy": "Eq(64*k**6 + 32*q*k**4 + 4*(q**2 - 4*s)*k**2 - r**2, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/resolvent-cubic", "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-bf5a941106", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "53", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "k_1^2 + k_2^2 + k_3^2 = -\\tfrac{1}{2}q", "name": null, "statement": "The sum of the squares of the three roots of the Descartes cubic equals minus q/2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k_1", "meaning": "root of the cubic in k^2" }, { "unit": null, "symbol": "k_2", "meaning": "root of the cubic in k^2" }, { "unit": null, "symbol": "k_3", "meaning": "root of the cubic in k^2" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in the reduced quartic" } ], "sympy": "Eq(k_1**2 + k_2**2 + k_3**2, -q/2)", "physics": false, "states": [], "concepts": [ "concept/root-of-an-equation", "concept/sum-of-the-roots", "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-c008449efe", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "53", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "k_1^2 k_2^2 k_3^2 = \\frac{r^2}{64}", "name": null, "statement": "The product of the squares of the three k roots equals r squared over 64.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k_1", "meaning": "root of the cubic in k^2" }, { "unit": null, "symbol": "k_2", "meaning": "root of the cubic in k^2" }, { "unit": null, "symbol": "k_3", "meaning": "root of the cubic in k^2" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in the reduced quartic" } ], "sympy": "Eq(k_1**2*k_2**2*k_3**2, r**2/64)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root-of-an-equation", "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-52fe7e096e", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "53", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "k_1 k_2 k_3 = -\\frac{r}{8}", "name": null, "statement": "The signs of the square roots k1, k2, k3 must be chosen so that their product equals minus r/8.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "k_1", "meaning": "square root of a root of the cubic in k^2" }, { "unit": null, "symbol": "k_2", "meaning": "square root of a root of the cubic in k^2" }, { "unit": null, "symbol": "k_3", "meaning": "square root of a root of the cubic in k^2" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in the reduced quartic" } ], "sympy": "Eq(k_1*k_2*k_3, -r/8)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root", "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-412d4651ba", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "53", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "k_1 + k_2 + k_3", "name": null, "statement": "One of the four roots of the reduced quartic in the symmetric form of Descartes' solution, with signs chosen as in the relation for k1 k2 k3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k_1", "meaning": "square root of a root of the cubic in k^2" }, { "unit": null, "symbol": "k_2", "meaning": "square root of a root of the cubic in k^2" }, { "unit": null, "symbol": "k_3", "meaning": "square root of a root of the cubic in k^2" } ], "sympy": "Eq(k_1 + k_2 + k_3, k_1 + k_2 + k_3)", "physics": false, "states": [], "concepts": [ "concept/root-of-an-equation", "concept/sum", "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-42f45803e8", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "53", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "64y^3 + 32qy^2 + 4(q^2 - 4s)y - r^2 = 0", "name": null, "statement": "The cubic whose roots are k1^2, k2^2, k3^2, used in the symmetric form of Descartes' solution.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown, equal to k^2" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in the reduced quartic" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in the reduced quartic" }, { "unit": null, "symbol": "s", "meaning": "constant term of the reduced quartic" } ], "sympy": "Eq(64*y**3 + 32*q*y**2 + 4*(q**2 - 4*s)*y - r**2, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/root-of-an-equation", "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-61a93dd16a", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "53", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "z = \\sqrt{y_1} + \\sqrt{y_2} + \\sqrt{y_3}", "name": null, "statement": "A root of the reduced quartic is the sum of the square roots of the three roots of the cubic in y, with suitable signs.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown of the reduced quartic" }, { "unit": null, "symbol": "y_1", "meaning": "root of the cubic 64y^3 + ... = 0" }, { "unit": null, "symbol": "y_2", "meaning": "root of the cubic 64y^3 + ... = 0" }, { "unit": null, "symbol": "y_3", "meaning": "root of the cubic 64y^3 + ... = 0" } ], "sympy": "Eq(z, sqrt(y_1) + sqrt(y_2) + sqrt(y_3))", "physics": false, "states": [], "concepts": [ "concept/root", "concept/root-of-an-equation", "concept/sum", "method/descartes-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-9d740d680a", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\sqrt{y_1}·\\sqrt{y_2}·\\sqrt{y_3} = -\\frac{r}{8}", "name": null, "statement": "The square roots in the sum for z must be chosen so that their product equals minus r/8.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "root of the cubic in y" }, { "unit": null, "symbol": "y_2", "meaning": "root of the cubic in y" }, { "unit": null, "symbol": "y_3", "meaning": "root of the cubic in y" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in the reduced quartic" } ], "sympy": "Eq(sqrt(y_1)*sqrt(y_2)*sqrt(y_3), -r/8)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-786adabbfb", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "z^3 - \\tfrac{3}{4}z - \\tfrac{1}{4}\\cos 3A = 0 \\qquad (z = \\cos A)", "name": null, "statement": "The triple-angle identity rewritten as a cubic in z = cos A, which is the form used for the trigonometric solution.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "cos A" }, { "unit": "degree of angle", "symbol": "A", "meaning": "auxiliary angle" } ], "sympy": "Eq(z**3 - Rational(3,4)*z - cos(3*A)/4, 0)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cubic-equation", "method/trigonometric-solution-of-a-cubic" ] }, { "id": "dickson-theory-of-equations-1922/eq-cd6c867298", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\cos 3A = 4\\cos^3 A - 3\\cos A", "name": null, "statement": "The triple-angle identity for cosine, expressing cos 3A in terms of cos A.", "kind": "identity", "symbols": [ { "unit": "degree of angle", "symbol": "A", "meaning": "an angle" } ], "sympy": "Eq(cos(3*A), 4*cos(A)**3 - 3*cos(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/trigonometry", "theorem/trigonometric-identity" ] }, { "id": "dickson-theory-of-equations-1922/eq-312e8df0ad", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "n = \\sqrt{-\\tfrac{4}{3}p}", "name": null, "statement": "The scale factor n that makes the substitution y = nz produce the cosine form of the cubic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "scale factor in the substitution y = nz" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" } ], "sympy": "Eq(n, sqrt(-Rational(4,3)*p))", "physics": false, "states": [], "concepts": [ "concept/reduced-cubic-equation", "method/substitution", "method/trigonometric-solution-of-a-cubic" ] }, { "id": "dickson-theory-of-equations-1922/eq-5b68683450", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "\\cos{3A} = -\\tfrac{1}{2}q ÷ \\sqrt{-p^{3}/27}", "name": null, "statement": "The angle A is defined by cos 3A equal to minus q/2 divided by the square root of minus p cubed over 27.", "kind": "definition", "symbols": [ { "unit": "degree of angle", "symbol": "A", "meaning": "auxiliary angle" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" } ], "sympy": "Eq(cos(3*A), (-q/2)/sqrt(-p**3/27))", "physics": false, "states": [], "concepts": [ "concept/cosine", "method/division", "method/trigonometric-solution-of-a-cubic" ] }, { "id": "dickson-theory-of-equations-1922/eq-40e6fa3a96", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "z^3 + \\frac{p}{n^2}z + \\frac{q}{n^3} = 0", "name": null, "statement": "After the substitution y = nz the reduced cubic becomes this equation in z, to be matched with the triple-angle form.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "cos A in the trigonometric solution" }, { "unit": null, "symbol": "n", "meaning": "scale factor of the substitution y = nz" }, { "unit": null, "symbol": "p", "meaning": "coefficient of y in the reduced cubic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" } ], "sympy": "Eq(z**3 + p/n**2*z + q/n**3, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "method/substitution", "method/trigonometric-solution-of-a-cubic" ] }, { "id": "dickson-theory-of-equations-1922/eq-6b7c2caf8a", "chapter": "dickson-theory-of-equations-1922/ch-iv", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Solution of Cubic and Quartic Equations; Their Discriminants", "latex": "R = -\\Delta/108", "name": null, "statement": "In the irreducible case the quantity R equals minus the discriminant divided by 108.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "quantity (p/3)^3 + (q/2)^2" }, { "unit": null, "symbol": "Delta", "meaning": "discriminant of the cubic" } ], "sympy": "Eq(R, -Delta/108)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/discriminant", "concept/irreducible-case" ] }, { "id": "dickson-theory-of-equations-1922/eq-db5d67deec", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "55", "location": "The Graph of an Equation", "latex": "x^2 - 6x - 3 = 0", "name": null, "statement": "The quadratic equation whose real roots are found graphically as the example of Use of Graphs.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown, the abscissa of a point on the graph" } ], "sympy": "Eq(x**2 - 6*x - 3, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation", "method/graphical-solution-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-6b933c787c", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "55", "location": "The Graph of an Equation", "latex": "y = x^2 - 6x - 3", "name": null, "statement": "The equation of the graph (a parabola) whose intersections with the x-axis give the roots of the quadratic equation.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the ordinate of a point on the graph, equal to the left member of the equation" }, { "unit": null, "symbol": "x", "meaning": "the abscissa of a point on the graph" } ], "sympy": "Eq(y, x**2 - 6*x - 3)", "physics": false, "states": [], "concepts": [ "concept/abscissa", "concept/function", "concept/graph-of-a-function", "concept/ordinate", "method/graphical-solution-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-1901272b31", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "55", "location": "The Graph of an Equation", "latex": "y = 8x^4 - 14x^3 - 9x^2 + 11x - 2", "name": null, "statement": "The quartic whose graph is used to illustrate that a curve sketched through integral points can mislead about the real roots.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the value of the polynomial at x" }, { "unit": null, "symbol": "x", "meaning": "the abscissa" } ], "sympy": "Eq(y, 8*x**4 - 14*x**3 - 9*x**2 + 11*x - 2)", "physics": false, "states": [], "concepts": [ "concept/graph-of-a-function", "concept/polynomial", "concept/real-root" ] }, { "id": "dickson-theory-of-equations-1922/eq-6758a72b9a", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "56", "location": "The Graph of an Equation", "latex": "8x^4 - 14x^3 - 9x^2 + 11x - 2 = 0", "name": null, "statement": "The quartic equation whose real roots are -1, 2, 1/4 and 1/2 as shown by the full graph.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown" } ], "sympy": "Eq(8*x**4 - 14*x**3 - 9*x**2 + 11*x - 2, 0)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/real-root", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-fdf49d6e1c", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "56", "location": "The Graph of an Equation", "latex": "y = x^3 + 4x^2 - 11", "name": null, "statement": "The cubic whose graph crosses the x-axis only once.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the ordinate of the graph" }, { "unit": null, "symbol": "x", "meaning": "the abscissa" } ], "sympy": "Eq(y, x**3 + 4*x**2 - 11)", "physics": false, "states": [], "concepts": [ "concept/graph-of-a-function", "concept/polynomial", "concept/real-root" ] }, { "id": "dickson-theory-of-equations-1922/eq-f8f5dc6e67", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "57", "location": "The Graph of an Equation", "latex": "\\frac{Y - y}{h} = \\frac{f(x+h) - f(x)}{h}", "name": null, "statement": "The slope of the secant PQ of the graph is the difference quotient of f over the increment h.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the polynomial whose graph is drawn" }, { "unit": null, "symbol": "h", "meaning": "the increment of x from P to Q" }, { "unit": null, "symbol": "x", "meaning": "abscissa of P" }, { "unit": null, "symbol": "y", "meaning": "ordinate of P" }, { "unit": null, "symbol": "Y", "meaning": "ordinate of Q" } ], "sympy": "Eq((Y - y)/h, (f(x + h) - f(x))/h)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/difference-quotient", "concept/limit", "concept/slope-of-a-curve" ] }, { "id": "dickson-theory-of-equations-1922/eq-c409af3e74", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "58", "location": "The Graph of an Equation", "latex": "f(x) = a_0 x^n + a_1 x^{n-1} + \\dotsb + a_{n-1} x + a_n", "name": null, "statement": "The general polynomial of degree n with real coefficients a_0 through a_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "the polynomial" }, { "unit": null, "symbol": "n", "meaning": "the degree of the polynomial" }, { "unit": null, "symbol": "a_0, ..., a_n", "meaning": "the coefficients" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/function", "concept/polynomial", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-21a5ab2501", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "The Graph of an Equation", "latex": "f'(x) = na_0 x^{n-1} + (n-1)a_1 x^{n-2} + \\dotsb + 2a_{n-2} x + a_{n-1}", "name": null, "statement": "The derivative of the polynomial f, obtained by multiplying each term by its exponent and lowering the exponent by one.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f'(x)", "meaning": "the first derivative of f" }, { "unit": null, "symbol": "n", "meaning": "degree of f" }, { "unit": null, "symbol": "a_i", "meaning": "coefficients of f" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/polynomial", "theorem/power-rule" ] }, { "id": "dickson-theory-of-equations-1922/eq-693f81b1fb", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "The Graph of an Equation", "latex": "f''(x) = n(n-1)a_0 x^{n-2} + (n-1)(n-2)a_1 x^{n-3} + \\dotsb + 2a_{n-2}", "name": null, "statement": "The second derivative of the polynomial f, the derivative of f'.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f''(x)", "meaning": "the second derivative of f" }, { "unit": null, "symbol": "n", "meaning": "degree of f" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/higher-order-derivative" ] }, { "id": "dickson-theory-of-equations-1922/eq-966d1ba68d", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "The Graph of an Equation", "latex": "0! = 1", "name": "zero factorial", "statement": "By definition the factorial of zero is one.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "r!", "meaning": "r factorial, the product 1·2·3···r" } ], "sympy": "Eq(factorial(0), 1)", "physics": false, "states": [ "theorem/zero-factorial" ], "concepts": [ "concept/factorial" ] }, { "id": "dickson-theory-of-equations-1922/eq-647b90521e", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "The Graph of an Equation", "latex": "y = f(\\alpha) + f'(\\alpha)(x-\\alpha)", "name": null, "statement": "The equation of the tangent to the graph of y = f(x) at the point with abscissa alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "abscissa of the point of tangency" }, { "unit": null, "symbol": "f'(alpha)", "meaning": "slope of the graph at alpha" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/slope-of-a-curve", "concept/tangent" ] }, { "id": "dickson-theory-of-equations-1922/eq-8dd67f90f9", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "The Graph of an Equation", "latex": "y-\\beta = s(x -\\alpha)", "name": null, "statement": "The equation of the straight line through the point (alpha, beta) with slope s.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s", "meaning": "slope of the line" }, { "unit": null, "symbol": "(alpha, beta)", "meaning": "a point on the line" } ], "sympy": "Eq(y - beta, s*(x - alpha))", "physics": false, "states": [], "concepts": [ "concept/line", "concept/slope-of-a-curve" ] }, { "id": "dickson-theory-of-equations-1922/eq-2961943cc5", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "63", "location": "The Graph of an Equation", "latex": "\\tan\\theta=f'(\\alpha)", "name": null, "statement": "The angle theta between the tangent and the X-axis has tangent equal to the slope f'(alpha).", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "angle between the tangent and the X-axis" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/slope-of-a-curve", "concept/tangent-function", "quantity/angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-1ec24a4431", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "63", "location": "The Graph of an Equation", "latex": "X = x\\cos\\theta", "name": null, "statement": "Relation between the old oblique-coordinate abscissa X and the new coordinate x, with the tangent angle theta.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "X", "meaning": "abscissa in the first system of axes" }, { "unit": null, "symbol": "x", "meaning": "new abscissa along the tangent" }, { "unit": null, "symbol": "theta", "meaning": "angle between the tangent and the X-axis" } ], "sympy": "Eq(X, x*cos(theta))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/transformation", "quantity/angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-0244af2213", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "63", "location": "The Graph of an Equation", "latex": "Y = f'(\\alpha)X + f''(\\alpha)\\frac{X^2}{2} + \\dotsb", "name": null, "statement": "The graph near the point of tangency, referred to axes parallel to the old ones through (alpha, beta), expanded in powers of X.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Y", "meaning": "ordinate relative to (alpha, beta)" }, { "unit": null, "symbol": "X", "meaning": "abscissa relative to (alpha, beta)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/tangent", "concept/transformation" ] }, { "id": "dickson-theory-of-equations-1922/eq-ad9fce4e94", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "63", "location": "The Graph of an Equation", "latex": "y = cx^m + dx^{m+1} + \\dotsb", "name": null, "statement": "The curve near its point of tangency, in oblique axes with the tangent as x-axis, begins with the term c x^m.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "the order of contact of the tangent (multiplicity)" }, { "unit": null, "symbol": "c", "meaning": "leading coefficient" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inflection-point", "concept/ordinary-tangent", "concept/tangent" ] }, { "id": "dickson-theory-of-equations-1922/eq-7d5977b7db", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "63", "location": "The Graph of an Equation", "latex": "c = \\frac{f^{(m)}(\\alpha)\\cos^m \\theta}{m!}", "name": null, "statement": "The leading coefficient c of the curve relative to the tangent, nonzero when f^(m)(alpha) is nonzero.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "leading coefficient in oblique axes" }, { "unit": null, "symbol": "m", "meaning": "multiplicity of the tangency" }, { "unit": null, "symbol": "theta", "meaning": "angle of the tangent" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/factorial", "concept/higher-order-derivative" ] }, { "id": "dickson-theory-of-equations-1922/eq-39c6da23aa", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "65", "location": "The Graph of an Equation", "latex": "f(x) = x^3 - 3lx + q", "name": null, "statement": "The reduced real cubic whose real roots are classified by the sign of q^2 - 4l^3.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "l", "meaning": "a constant with l not zero" }, { "unit": null, "symbol": "q", "meaning": "a constant term" } ], "sympy": "Eq(f(x), x**3 - 3*l*x + q)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/polynomial", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-32a16d1e96", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "65", "location": "The Graph of an Equation", "latex": "q^2 = 4l^3", "name": null, "statement": "The condition under which a bend point of the reduced cubic lies on the x-axis, so that the cubic has a double root.", "kind": "result", "symbols": [ { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" }, { "unit": null, "symbol": "l", "meaning": "constant of the reduced cubic" } ], "sympy": "Eq(q**2, 4*l**3)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/discriminant", "concept/multiple-root" ] }, { "id": "dickson-theory-of-equations-1922/eq-d30219e428", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "65", "location": "The Graph of an Equation", "latex": "q^2 < 4l^3", "name": null, "statement": "The condition under which the reduced cubic has three distinct real roots.", "kind": "result", "symbols": [ { "unit": null, "symbol": "q", "meaning": "constant term of the reduced cubic" }, { "unit": null, "symbol": "l", "meaning": "constant of the reduced cubic" } ], "sympy": "Lt(q**2, 4*l**3)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/discriminant", "concept/real-root" ] }, { "id": "dickson-theory-of-equations-1922/eq-8b024973a0", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "The Graph of an Equation", "latex": "D = f(a+h) - f(a)", "name": null, "statement": "The difference whose smallness as h tends to zero defines continuity of f at a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "D", "meaning": "the difference f(a+h) - f(a)" }, { "unit": null, "symbol": "a", "meaning": "a real constant" }, { "unit": null, "symbol": "h", "meaning": "a small real increment" } ], "sympy": "Eq(D, f(a + h) - f(a))", "physics": false, "states": [], "concepts": [ "concept/continuity", "concept/continuous-function", "concept/difference", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-25e92bf677", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "The Graph of an Equation", "latex": "F = a_1 h + a_2 h^2 + \\dotsb + a_n h^n", "name": null, "statement": "The polynomial in h with no constant term, shown to be numerically small for small h.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "F", "meaning": "polynomial in h without constant term" }, { "unit": null, "symbol": "a_i", "meaning": "real coefficients" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuity", "concept/continuous-function", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-166d10edce", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "67", "location": "The Graph of an Equation", "latex": "k < \\frac{p}{p + g}", "name": null, "statement": "The bound on |h| that guarantees the polynomial's increment is less than p.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "k", "meaning": "bound on |h|, with k < 1" }, { "unit": null, "symbol": "p", "meaning": "an assigned positive number" }, { "unit": null, "symbol": "g", "meaning": "greatest numerical value of the coefficients a_1 through a_n" } ], "sympy": "Lt(k, p/(p + g))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/continuity", "concept/continuous-function" ] }, { "id": "dickson-theory-of-equations-1922/eq-2f7718e15f", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "68", "location": "The Graph of an Equation", "latex": "f(x) = x^n (a_0 + \\phi)", "name": null, "statement": "Factoring x^n out of the polynomial shows that for large x its sign is that of a_0 x^n.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the sum a_1/x + ... + a_n/x^n" }, { "unit": null, "symbol": "a_0", "meaning": "leading coefficient" } ], "sympy": "Eq(f(x), x**n*(a0 + phi))", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/sign-of-a-polynomial-at-infinity", "concept/sufficiently-large-values" ] }, { "id": "dickson-theory-of-equations-1922/eq-db7be62b8a", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "68", "location": "The Graph of an Equation", "latex": "\\phi = \\frac{a_1}{x} + \\frac{a_2}{x^2} + \\dotsb + \\frac{a_n}{x^n}", "name": null, "statement": "The quantity phi collects the lower-order terms divided by powers of x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "remainder of the polynomial divided by x^n and a_0 aside" }, { "unit": null, "symbol": "a_i", "meaning": "coefficients" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/polynomial", "concept/sign-of-a-polynomial-at-infinity" ] }, { "id": "dickson-theory-of-equations-1922/eq-cacc5c228e", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "69", "location": "The Graph of an Equation", "latex": "\\frac{(x-a)(x-b)f'(x)}{f(x)} \\equiv r(x-b) + s(x-a) + (x-a)(x-b) \\frac{Q'(x)}{Q(x)}", "name": null, "statement": "The logarithmic-derivative identity for f = (x-a)^r (x-b)^s Q(x), used in the proof of Rolle's theorem.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the polynomial with roots a and b" }, { "unit": null, "symbol": "r", "meaning": "multiplicity of root a" }, { "unit": null, "symbol": "s", "meaning": "multiplicity of root b" }, { "unit": null, "symbol": "Q", "meaning": "polynomial divisible by neither x-a nor x-b" } ], "sympy": "Eq((x-a)*(x-b)*Derivative(f(x), x)/f(x), r*(x-b) + s*(x-a) + (x-a)*(x-b)*Derivative(Q(x), x)/Q(x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/multiple-root", "theorem/product-rule", "theorem/rolle-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/eq-7c20e94c81", "chapter": "dickson-theory-of-equations-1922/ch-v", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "70", "location": "The Graph of an Equation", "latex": "\\tfrac{1}{15}f'(x) = x^4 - 5x^2 + 4 = (x^2 - 1)(x^2 - 4)", "name": null, "statement": "Dividing the derivative of 3x^5 - 25x^3 + 60x - 20 by 15 gives a quartic that factors into (x^2 - 1)(x^2 - 4), locating the roots of f'(x) = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "the polynomial 3x^5 - 25x^3 + 60x - 20" } ], "sympy": "Eq(Rational(1, 15)*Derivative(f(x), x), x**4 - 5*x**2 + 4)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/factor", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-bff8156735", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "72", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f(x) \\equiv a_0 x^n + a_1 x^{n-1} + \\dotsb + a_l x^{n-l}", "name": null, "statement": "Any real polynomial f(x) is written as a sum of descending powers of x with real coefficients, where the leading coefficient a_0 and the coefficient a_l are nonzero.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "real polynomial in x" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" }, { "unit": null, "symbol": "a_0", "meaning": "leading coefficient" }, { "unit": null, "symbol": "a_l", "meaning": "coefficient of the last term written (a_l nonzero)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/degree", "concept/leading-coefficient", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-a2fcb3e3ba", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "72", "location": "Isolation of the Real Roots of a Real Equation", "latex": "F(x) \\equiv (x-r)f(x) \\equiv A_0 x^{n+1} + A_1 x^n + \\dotsb + A_{l+1}x^{n-l}", "name": null, "statement": "Multiplying f(x) by (x - r) gives a polynomial F(x) whose coefficients A_i are written in terms of the coefficients a_i of f.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "F(x)", "meaning": "product (x - r) f(x)" }, { "unit": null, "symbol": "f(x)", "meaning": "the polynomial being multiplied" }, { "unit": null, "symbol": "r", "meaning": "a positive real number" }, { "unit": null, "symbol": "A_i", "meaning": "coefficients of F(x)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/positive-number", "concept/product", "concept/variation-of-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-b3b77d9f0f", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "72", "location": "Isolation of the Real Roots of a Real Equation", "latex": "A_1 = a_1 - ra_0", "name": null, "statement": "The second coefficient of F(x) equals a_1 minus r times a_0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A_1", "meaning": "second coefficient of F(x) = (x - r) f(x)" }, { "unit": null, "symbol": "a_1", "meaning": "second coefficient of f(x)" }, { "unit": null, "symbol": "a_0", "meaning": "leading coefficient of f(x)" }, { "unit": null, "symbol": "r", "meaning": "a positive real number" } ], "sympy": "Eq(A_1, a_1 - r*a_0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/product" ] }, { "id": "dickson-theory-of-equations-1922/eq-1eccab8397", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "72", "location": "Isolation of the Real Roots of a Real Equation", "latex": "A_{l+1} = -r a_l", "name": null, "statement": "The last coefficient of F(x) equals minus r times the last coefficient a_l of f(x).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A_{l+1}", "meaning": "last coefficient of F(x)" }, { "unit": null, "symbol": "a_l", "meaning": "last coefficient of f(x)" }, { "unit": null, "symbol": "r", "meaning": "a positive real number" } ], "sympy": "Eq(A_lp1, -r*a_l)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/product" ] }, { "id": "dickson-theory-of-equations-1922/eq-ec76e989a8", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "73", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f(x) \\equiv (x - r_1)\\dotsm (x - r_k)\\phi(x)", "name": null, "statement": "A polynomial whose positive real roots are r_1 to r_k factors as the product of the linear factors (x - r_i) and a remaining polynomial phi(x).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "polynomial with positive real roots r_1,...,r_k and no others" }, { "unit": null, "symbol": "r_i", "meaning": "positive real roots (not necessarily distinct)" }, { "unit": null, "symbol": "phi(x)", "meaning": "polynomial with real coefficients having no positive real roots" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/factor", "concept/multiple-root", "concept/polynomial", "concept/positive-number", "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-02a3690bff", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "76", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f = q_1 f_1 - f_2", "name": null, "statement": "Dividing f by its derivative f_1 gives quotient q_1 and the negative of the remainder, which is the second function f_2 in Sturm's chain.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the polynomial whose real roots are isolated" }, { "unit": null, "symbol": "f_1", "meaning": "the first derivative of f" }, { "unit": null, "symbol": "q_1", "meaning": "quotient of f divided by f_1" }, { "unit": null, "symbol": "f_2", "meaning": "negative of the remainder of f divided by f_1" } ], "sympy": "Eq(f, q_1*f_1 - f_2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/polynomial", "concept/quotient", "concept/remainder", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-3d4b8d70fb", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "77", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f_{i-1}(x) = q_i f_i(x) - f_{i+1}(x)", "name": null, "statement": "Each Sturm function is the quotient times the next Sturm function minus the one after it, the general step of the Sturm chain.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f_{i-1}(x)", "meaning": "Sturm function preceding f_i" }, { "unit": null, "symbol": "f_i(x)", "meaning": "Sturm function" }, { "unit": null, "symbol": "f_{i+1}(x)", "meaning": "Sturm function following f_i (negative of a remainder)" }, { "unit": null, "symbol": "q_i", "meaning": "quotient at step i" } ], "sympy": "Eq(f_im1(x), q_i*f_i(x) - f_ip1(x))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/quotient", "concept/remainder", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-4ece3b817e", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "77", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f_{i-1}(\\rho) = -f_{i+1}(\\rho) \\ne 0", "name": null, "statement": "At a root rho of f_i, the neighbouring Sturm functions f_{i-1} and f_{i+1} take equal and opposite values, and that value is not zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "a root of f_i(x) = 0" }, { "unit": null, "symbol": "f_{i-1}", "meaning": "Sturm function preceding f_i" }, { "unit": null, "symbol": "f_{i+1}", "meaning": "Sturm function following f_i" } ], "sympy": "Eq(f_im1(rho), -f_ip1(rho))", "physics": false, "states": [], "concepts": [ "concept/function-value", "concept/root", "concept/sturm-s-functions", "concept/variation-of-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-dce102a460", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "78", "location": "Isolation of the Real Roots of a Real Equation", "latex": "V_{r-p} - V_{r+p} = 1", "name": null, "statement": "Passing a simple root r of f from left to right lowers the number of variations of sign of the Sturm sequence by exactly one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V_x", "meaning": "number of variations of sign of the Sturm sequence f, f_1, ..., f_n at x" }, { "unit": null, "symbol": "r", "meaning": "a root of f(x) = 0" }, { "unit": null, "symbol": "p", "meaning": "a sufficiently small positive number" } ], "sympy": "Eq(V_minus - V_plus, 1)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/root", "concept/sturm-s-functions", "concept/variation-of-sign", "theorem/sturm-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/eq-688575c9b0", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "78", "location": "Isolation of the Real Roots of a Real Equation", "latex": "V_a\\geqq V_{b}", "name": null, "statement": "If a is less than b, the number of variations of sign at a is at least the number at b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V_a", "meaning": "number of variations of sign of the Sturm sequence at x = a" }, { "unit": null, "symbol": "V_b", "meaning": "number of variations of sign of the Sturm sequence at x = b" }, { "unit": null, "symbol": "a", "meaning": "a real number less than b" }, { "unit": null, "symbol": "b", "meaning": "a real number greater than a" } ], "sympy": "Ge(V_a, V_b)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/variation-of-sign", "theorem/sturm-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/eq-22c1a91c9f", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Isolation of the Real Roots of a Real Equation", "latex": "c_{i+1} F_{i-1}(\\rho) = -k_{i+1}(\\rho) F_{i+1}(\\rho)", "name": null, "statement": "With the modified Sturm functions, at a root rho of F_i, F_{i-1} and F_{i+1} have opposite signs because their values are related by positive constants and a positive factor.", "kind": "result", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "a root of F_i(x) = 0" }, { "unit": null, "symbol": "c_{i+1}", "meaning": "positive constant" }, { "unit": null, "symbol": "F_{i-1}", "meaning": "modified Sturm function preceding F_i" }, { "unit": null, "symbol": "F_{i+1}", "meaning": "modified Sturm function following F_i" }, { "unit": null, "symbol": "k_{i+1}", "meaning": "positive factor removed from the Sturm function" } ], "sympy": "Eq(c_{i+1}*F_{i-1}(rho), -k_{i+1}(rho)*F_{i+1}(rho))", "physics": false, "states": [], "concepts": [ "concept/positive-number", "concept/root", "concept/sturm-s-functions", "concept/variation-of-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-777063e486", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f = z^4 + qz^2 + rz + s", "name": null, "statement": "The reduced quartic is defined with no cubic term, its coefficients named q, r, s.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f", "meaning": "reduced quartic polynomial in z" }, { "unit": null, "symbol": "z", "meaning": "variable" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z" }, { "unit": null, "symbol": "s", "meaning": "constant term" } ], "sympy": "Eq(f, z**4 + q*z**2 + r*z + s)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/quartic-equation", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-f0f118b2a3", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f_1 = 4z^3 + 2qz + r", "name": null, "statement": "The first derivative of the reduced quartic f with respect to z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f_1", "meaning": "first derivative of f with respect to z" }, { "unit": null, "symbol": "z", "meaning": "variable" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in f" } ], "sympy": "Eq(f_1, 4*z**3 + 2*q*z + r)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/reduced-quartic-equation", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-1752c87f49", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f_2 = -2qz^2 - 3rz - 4s", "name": null, "statement": "The second Sturm function of the reduced quartic, the negative of the remainder of f divided by f_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f_2", "meaning": "second Sturm function of the reduced quartic" }, { "unit": null, "symbol": "z", "meaning": "variable" }, { "unit": null, "symbol": "q", "meaning": "coefficient in f" }, { "unit": null, "symbol": "r", "meaning": "coefficient in f" }, { "unit": null, "symbol": "s", "meaning": "constant term of f" } ], "sympy": "Eq(f_2, -2*q*z**2 - 3*r*z - 4*s)", "physics": false, "states": [], "concepts": [ "concept/reduced-quartic-equation", "concept/remainder", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-d66f92d186", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f_3 = Lz - 12rs - rq^2", "name": null, "statement": "The third Sturm function of the reduced quartic is linear in z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f_3", "meaning": "third Sturm function of the reduced quartic" }, { "unit": null, "symbol": "L", "meaning": "auxiliary quantity 8qs - 2q^3 - 9r^2" }, { "unit": null, "symbol": "z", "meaning": "variable" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in f" }, { "unit": null, "symbol": "s", "meaning": "constant term of f" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" } ], "sympy": "Eq(f_3, L*z - 12*r*s - r*q**2)", "physics": false, "states": [], "concepts": [ "concept/reduced-quartic-equation", "concept/remainder", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-4e68fe907a", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "L = 8qs - 2q^3 - 9r^2", "name": null, "statement": "L is defined as 8qs minus 2q cubed minus 9r squared.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "L", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in f" }, { "unit": null, "symbol": "s", "meaning": "constant term of f" } ], "sympy": "Eq(L, 8*q*s - 2*q**3 - 9*r**2)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-1eb93a9c1c", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "\\Delta = -4P^3 - 27Q^2", "name": null, "statement": "The discriminant of the reduced quartic is expressed through the auxiliary quantities P and Q.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "discriminant of the reduced quartic" }, { "unit": null, "symbol": "P", "meaning": "auxiliary quantity -4s - q^2/3" }, { "unit": null, "symbol": "Q", "meaning": "auxiliary quantity 8qs/3 - r^2 - 2q^3/27" } ], "sympy": "Eq(Delta, -4*P**3 - 27*Q**2)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/quartic-equation", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-b8b56945d9", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "P = -4s - \\frac{q^2}{3}", "name": null, "statement": "P is defined as minus 4s minus q squared over 3.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "P", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "s", "meaning": "constant term of f" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" } ], "sympy": "Eq(P, -4*s - q**2/3)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-bc6db44bd5", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "Q = \\tfrac{8}{3}qs - r^2 - \\tfrac{2}{27}q^3", "name": null, "statement": "Q is defined in terms of the coefficients q, r, s of the reduced quartic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in f" }, { "unit": null, "symbol": "s", "meaning": "constant term of f" } ], "sympy": "Eq(Q, Rational(8,3)*q*s - r**2 - Rational(2,27)*q**3)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-34c02cbed8", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "4s = -P - \\frac{q^2}{3}", "name": null, "statement": "The constant term s can be eliminated: 4s equals minus P minus q squared over 3.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "s", "meaning": "constant term of f" }, { "unit": null, "symbol": "P", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" } ], "sympy": "Eq(4*s, -P - q**2/3)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/discriminant", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-a16b8fc2d7", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "r^2 = -Q - \\tfrac{2}{3}qP - \\tfrac{8}{27}q^3", "name": null, "statement": "The square of the coefficient r can be written in terms of Q, P and q.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "r", "meaning": "coefficient of z in f" }, { "unit": null, "symbol": "Q", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "P", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" } ], "sympy": "Eq(r**2, -Q - Rational(2,3)*q*P - Rational(8,27)*q**3)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/discriminant", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-7091be904d", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f_3 = Lz + 3rP", "name": null, "statement": "The third Sturm function of the reduced quartic written with the alternative auxiliary quantities P and L.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f_3", "meaning": "third Sturm function of the reduced quartic" }, { "unit": null, "symbol": "L", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "z", "meaning": "variable" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in f" }, { "unit": null, "symbol": "P", "meaning": "auxiliary quantity of the reduced quartic" } ], "sympy": "Eq(f_3, L*z + 3*r*P)", "physics": false, "states": [], "concepts": [ "concept/reduced-quartic-equation", "concept/remainder", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-185a40dbe0", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "L = 9Q + 4qP", "name": null, "statement": "L equals 9Q plus 4qP in terms of the auxiliary quantities Q and P.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "L", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "Q", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "P", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" } ], "sympy": "Eq(L, 9*Q + 4*q*P)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/reduced-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-03da7a5a48", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "18r^2 qP^2 - 9r^2 LP + 4sL^2 = q^2 \\Delta", "name": null, "statement": "The negative of the remainder of L^2 f_2 on division by f_3 equals q^2 times the discriminant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "discriminant of the reduced quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient of z^2 in f" }, { "unit": null, "symbol": "r", "meaning": "coefficient of z in f" }, { "unit": null, "symbol": "s", "meaning": "constant term of f" }, { "unit": null, "symbol": "L", "meaning": "auxiliary quantity of the reduced quartic" }, { "unit": null, "symbol": "P", "meaning": "auxiliary quantity of the reduced quartic" } ], "sympy": "Eq(18*r**2*q*P**2 - 9*r**2*L*P + 4*s*L**2, q**2*Delta)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/reduced-quartic-equation", "concept/remainder", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-e3d047fc4d", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "80", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f_4 = \\Delta", "name": null, "statement": "The last Sturm function of the reduced quartic is the (constant) discriminant, so the chain ends with the discriminant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f_4", "meaning": "fourth Sturm function of the reduced quartic, a constant" }, { "unit": null, "symbol": "Delta", "meaning": "discriminant of the reduced quartic" } ], "sympy": "Eq(f_4, Delta)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/discriminant", "concept/reduced-quartic-equation", "concept/sturm-s-functions" ] }, { "id": "dickson-theory-of-equations-1922/eq-50e7b4244d", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Isolation of the Real Roots of a Real Equation", "latex": "V_0 - V_{\\infty} = V", "name": null, "statement": "The number of variations of sign of f at x = 0 minus that at infinity equals V, the number of variations of sign of f(x), which gives Descartes' rule.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V_0", "meaning": "number of variations of sign of the Budan sequence at x = 0" }, { "unit": null, "symbol": "V_infty", "meaning": "number of variations of sign of the Budan sequence at x = +infinity" }, { "unit": null, "symbol": "V", "meaning": "number of variations of sign of f(x)" } ], "sympy": "Eq(V_0 - V_inf, V)", "physics": false, "states": [], "concepts": [ "concept/positive-number", "concept/variation-of-sign", "theorem/budan-s-theorem", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/eq-39cd545676", "chapter": "dickson-theory-of-equations-1922/ch-vi", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Isolation of the Real Roots of a Real Equation", "latex": "f(x) \\equiv a_0 x^n + a_1 x^{n-1} + \\dotsb + a_{n-1}x + a_n = 0", "name": null, "statement": "A general real polynomial equation of degree n with real coefficients and a_n nonzero, written with the constant term a_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "real polynomial in x" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" }, { "unit": null, "symbol": "a_0", "meaning": "leading coefficient" }, { "unit": null, "symbol": "a_n", "meaning": "constant term (nonzero)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant-term", "concept/degree", "concept/equation", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-341c4382d1", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "86", "location": "Solution of Numerical Equations", "latex": "p^3 + 6p^2 + 10p - 1 = 0", "name": null, "statement": "The transformed equation for p, obtained by putting x = 2 + p in x^3 - 2x - 5 = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "the shift x - 2, so that x = 2 + p" } ], "sympy": "Eq(p**3 + 6*p**2 + 10*p - 1, 0)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/root-of-an-equation", "concept/transformed-equation", "method/horner-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-070613b1b8", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "86", "location": "Solution of Numerical Equations", "latex": "x^3 - 2x - 5 \\equiv (x-2)^3 + 6(x-2)^2 + 10(x-2) - 1", "name": null, "statement": "An identity in x expressing the cubic x^3 - 2x - 5 in powers of x - 2, whose constant term -1 is the remainder on division by x - 2.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable of the polynomial" } ], "sympy": "Eq(x**3 - 2*x - 5, (x-2)**3 + 6*(x-2)**2 + 10*(x-2) - 1)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/remainder", "concept/transformed-equation", "method/horner-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-d49407fce0", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "87", "location": "Solution of Numerical Equations", "latex": "t^3 + 6.282t^2 + 11.154508t - 0.006153416 = 0", "name": null, "statement": "The transformed equation for the correction t to the root 2.094 of x^3 - 2x - 5 = 0, obtained by Horner's method.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "the correction added to 2.094 to give the root x" } ], "sympy": "Eq(t**3 + 6.282*t**2 + 11.154508*t - 0.006153416, 0)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/root-of-an-equation", "concept/transformed-equation", "method/horner-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-34a5c32978", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "91", "location": "Solution of Numerical Equations", "latex": "f(a+h) = f(a) + f'(a)h + f''(a) \\frac{h^2}{2} + \\dotsb", "name": "Taylor's formula", "statement": "Taylor's expansion of f(a+h) about a, with the higher powers of h indicated by dots.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "f", "meaning": "a function of x" }, { "unit": null, "symbol": "a", "meaning": "an approximate value of a real root" }, { "unit": null, "symbol": "h", "meaning": "a small correction to a" }, { "unit": null, "symbol": "f'(a)", "meaning": "derivative of f at a" }, { "unit": null, "symbol": "f''(a)", "meaning": "second derivative of f at a" } ], "sympy": null, "physics": false, "states": [ "method/taylor-s-theorem" ], "concepts": [ "concept/derivative", "concept/function", "concept/higher-order-derivative", "method/newton-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-31c3fd2191", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "91", "location": "Solution of Numerical Equations", "latex": "f(a) + f'(a)h = 0", "name": null, "statement": "Neglecting the powers h^2, h^3, ... of the small correction h leaves this linear condition on h.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f(a)", "meaning": "value of the function at the approximate root a" }, { "unit": null, "symbol": "f'(a)", "meaning": "derivative of the function at a" }, { "unit": null, "symbol": "h", "meaning": "correction to the approximate root a" } ], "sympy": "Eq(f(a) + Derivative(f(a), a)*h, 0)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/derivative", "concept/root-of-an-equation", "method/newton-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-64b87a678e", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "91", "location": "Solution of Numerical Equations", "latex": "h = \\frac{-f(a)}{f'(a)}", "name": "Newton's correction", "statement": "Newton's method takes the correction to the approximate root a as minus f(a) divided by f'(a); a + h is the next approximation.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "h", "meaning": "correction to the approximate root a" }, { "unit": null, "symbol": "a", "meaning": "current approximate value of a real root" }, { "unit": null, "symbol": "f", "meaning": "the function whose root is sought" } ], "sympy": "Eq(h, -f(a)/Derivative(f(a), a))", "physics": false, "states": [ "theorem/newton-s-correction" ], "concepts": [ "concept/approximation", "concept/derivative", "concept/root-of-an-equation", "method/newton-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-78e020cb16", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "92", "location": "Solution of Numerical Equations", "latex": "f'(a) = \\tan XTP", "name": null, "statement": "Geometrically the derivative at a point equals the tangent of the angle the tangent line makes with the x-axis, the tangent being drawn at P on the graph.", "kind": "law", "symbols": [ { "unit": null, "symbol": "f'(a)", "meaning": "slope of the graph at the point with abscissa a" }, { "unit": null, "symbol": "XTP", "meaning": "angle between the x-axis and the tangent at P" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/graph-of-a-function", "concept/tangent", "quantity/angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-4f85f5694d", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "93", "location": "Solution of Numerical Equations", "latex": "c = \\frac{\\alpha f(\\beta) - \\beta f(\\alpha)}{f(\\beta) - f(\\alpha)}", "name": "regula falsi formula", "statement": "The abscissa c where the chord joining the points at alpha and beta meets the x-axis, lying between alpha and beta.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "abscissa of the intersection of the chord AB with the x-axis" }, { "unit": null, "symbol": "alpha", "meaning": "one endpoint of the interval containing the root" }, { "unit": null, "symbol": "beta", "meaning": "the other endpoint of the interval containing the root" }, { "unit": null, "symbol": "f", "meaning": "the function whose root is sought" } ], "sympy": "Eq(c, (alpha*f(beta) - beta*f(alpha))/(f(beta) - f(alpha)))", "physics": false, "states": [ "theorem/regula-falsi-formula" ], "concepts": [ "concept/graph-of-a-function", "concept/interval", "concept/root-of-an-equation", "method/regula-falsi" ] }, { "id": "dickson-theory-of-equations-1922/eq-b0ab76ea7d", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "93", "location": "Solution of Numerical Equations", "latex": "-f(\\alpha) : c - \\alpha = f(\\beta) : \\beta - c", "name": null, "statement": "By similar triangles, the chord AB cuts the x-axis at c so that the two segments on the axis are in the same ratio as the two ordinates.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c", "meaning": "abscissa where the chord AB meets the x-axis" }, { "unit": null, "symbol": "alpha", "meaning": "left endpoint of the interval" }, { "unit": null, "symbol": "beta", "meaning": "right endpoint of the interval" } ], "sympy": "Eq(-f(alpha)/(c - alpha), f(beta)/(beta - c))", "physics": false, "states": [], "concepts": [ "concept/graph-of-a-function", "concept/similar-triangles", "method/regula-falsi" ] }, { "id": "dickson-theory-of-equations-1922/eq-2bc1e14efe", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "94", "location": "Solution of Numerical Equations", "latex": "f'(x) = 4x^3 + 3x^2 - 6x - 1", "name": null, "statement": "The derivative of f(x) = x^4 + x^3 - 3x^2 - x - 4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f'(x)", "meaning": "first derivative of f with respect to x" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq(Derivative(x**4 + x**3 - 3*x**2 - x - 4, x), 4*x**3 + 3*x**2 - 6*x - 1)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-c3a3422621", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Solution of Numerical Equations", "latex": "d^4 + 9d^3 + 27d^2 + 31d + 6 = 0", "name": null, "statement": "The transformed equation for d obtained by putting x = 2 + d in x^4 + x^3 - 3x^2 - x - 4 = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "d", "meaning": "the shift x - 2 from the approximate root 2" } ], "sympy": "Eq(d**4 + 9*d**3 + 27*d**2 + 31*d + 6, 0)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/root-of-an-equation", "concept/transformed-equation", "method/horner-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-87ade65e4d", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "90", "location": "Solution of Numerical Equations", "latex": "6.3r^2 + 11.16196r + 0.000541708 = 0", "name": null, "statement": "Newton's transformed equation for the second correction r, after neglecting the cubic term in q.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "second correction to the root, with q = -0.0054 + r" } ], "sympy": "Eq(6.3*r**2 + 11.16196*r + 0.000541708, 0)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/transformed-equation", "method/newton-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-06f2dbb720", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "90", "location": "Solution of Numerical Equations", "latex": "q^3 + 6.3q^2 + 11.23q + 0.061 = 0", "name": null, "statement": "Newton's transformed equation for the correction q to the approximation 0.1 of p, from x = 2 + 0.1 + q.", "kind": "result", "symbols": [ { "unit": null, "symbol": "q", "meaning": "correction added to p = 0.1" } ], "sympy": "Eq(q**3 + 6.3*q**2 + 11.23*q + 0.061, 0)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/transformed-equation", "method/newton-s-method" ] }, { "id": "dickson-theory-of-equations-1922/eq-9bc7fc81a4", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Solution of Numerical Equations", "latex": "x - \\sin x = \\tfrac{1}{4} \\pi", "name": null, "statement": "The condition that the chord cuts off a segment of one-eighth the circle's area, reduced to an equation in the central angle x.", "kind": "result", "symbols": [ { "unit": "radian", "symbol": "x", "meaning": "angle at the centre of the circle subtended by the chord" } ], "sympy": "Eq(x - sin(x), pi/4)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/circular-segment", "concept/sine", "quantity/angle", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/eq-b09e3bab8b", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Solution of Numerical Equations", "latex": "\\tfrac{1}{2} r^2(x - \\sin x) = \\tfrac{1}{8} \\pi r^2", "name": null, "statement": "The area of a circular segment with central angle x equals one-eighth of the area of the circle.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the circle" }, { "unit": "radian", "symbol": "x", "meaning": "central angle of the segment, measured in radians" } ], "sympy": "Eq(Rational(1,2)*r**2*(x - sin(x)), Rational(1,8)*pi*r**2)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/circular-segment", "concept/radius", "concept/sine", "quantity/angle", "quantity/area", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/eq-711f7fe774", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "97", "location": "Solution of Numerical Equations", "latex": "h = \\frac{-f(a)}{f'(a)} = \\frac{-a + \\sin a + \\tfrac{1}{4} \\pi}{1 - \\cos a}", "name": null, "statement": "Newton's correction for the root of x - sin x - pi/4 = 0, written with f(a) = a - sin a - pi/4 and f'(a) = 1 - cos a.", "kind": "rule", "symbols": [ { "unit": "radian", "symbol": "a", "meaning": "current approximation to the central angle, in radians" }, { "unit": "radian", "symbol": "h", "meaning": "correction to a" } ], "sympy": "Eq(h, (-a + sin(a) + pi/4)/(1 - cos(a)))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/sine", "method/newton-s-method", "quantity/angle", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/eq-ec8c48a589", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Solution of Numerical Equations", "latex": "f'(x) = 2 - \\frac{M}{x}", "name": null, "statement": "Derivative of f(x) = 2x - log x - 7 where log x = M log_e x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "M", "meaning": "modulus for converting natural logarithms to common logarithms, 0.4343" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq(Derivative(2*x - log(x, 10) - 7, x), 2 - M/x)", "physics": false, "states": [], "concepts": [ "concept/common-logarithm", "concept/derivative", "concept/function", "concept/logarithm" ] }, { "id": "dickson-theory-of-equations-1922/eq-f8a5d1b4f4", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Solution of Numerical Equations", "latex": "\\log x = M \\log_e x", "name": null, "statement": "A common logarithm equals the natural logarithm multiplied by the modulus M, which is about 0.4343.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "M", "meaning": "modulus 0.4343, the conversion factor from natural to common logarithms" }, { "unit": null, "symbol": "x", "meaning": "positive number whose logarithm is taken" } ], "sympy": "Eq(log(x, 10), M*log(x))", "physics": false, "states": [], "concepts": [ "concept/base-of-a-logarithm-system", "concept/common-logarithm", "concept/logarithm" ] }, { "id": "dickson-theory-of-equations-1922/eq-9f4b8e5344", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "86", "location": "Solution of Numerical Equations", "latex": "x^3 - 2x - 5 = 0", "name": null, "statement": "The equation whose real root between 2 and 3 is computed by Horner's method, and the reference point for the transformed equations.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown" } ], "sympy": "Eq(x**3 - 2*x - 5, 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/eq-a06b690dff", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Solution of Numerical Equations", "latex": "f(x) - f''(x) \\frac{y^2}{1·2} + f''''(x) \\frac{y^4}{1·2·3·4} - \\dotsb = 0", "name": null, "statement": "The real part of the Taylor expansion of f(x+yi) set to zero, giving the first of the two real equations for an imaginary root.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the root" }, { "unit": null, "symbol": "y", "meaning": "imaginary part of the root" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/higher-order-derivative", "concept/imaginary-root", "method/taylor-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/eq-e3f521bfae", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Solution of Numerical Equations", "latex": "f'(x) - f'''(x) \\frac{y^2}{1·2·3} + f^{(5)}(x)\\frac{y^4}{5!} - \\dotsb = 0", "name": null, "statement": "The imaginary part of the Taylor expansion of f(x+yi) set to zero, giving the second real equation for an imaginary root.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the root" }, { "unit": null, "symbol": "y", "meaning": "imaginary part of the root" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/higher-order-derivative", "concept/imaginary-root", "method/taylor-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/eq-d2b9c97056", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Solution of Numerical Equations", "latex": "x^4 - x + 1 - 6x^2 y^2 + y^4 = 0", "name": null, "statement": "The real equation obtained from z^4 - z + 1 = 0 with z = x + yi, by setting the real part to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the root z" }, { "unit": null, "symbol": "y", "meaning": "imaginary part of the root z" } ], "sympy": "Eq(x**4 - x + 1 - 6*x**2*y**2 + y**4, 0)", "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-5e64076267", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Solution of Numerical Equations", "latex": "4x^3 - 1 - 4xy^2 = 0", "name": null, "statement": "The imaginary-part equation obtained from z^4 - z + 1 = 0 with z = x + yi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the root z" }, { "unit": null, "symbol": "y", "meaning": "imaginary part of the root z" } ], "sympy": "Eq(4*x**3 - 1 - 4*x*y**2, 0)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/imaginary-root", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-c437bf27d5", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Solution of Numerical Equations", "latex": "y^2 = x^2 - \\frac{1}{4x}", "name": null, "statement": "Solving the imaginary-part equation for y^2 in terms of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "imaginary part of the root z" }, { "unit": null, "symbol": "x", "meaning": "real part of the root z" } ], "sympy": "Eq(y**2, x**2 - 1/(4*x))", "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-feb66f0bea", "chapter": "dickson-theory-of-equations-1922/ch-vii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Solution of Numerical Equations", "latex": "-4x^6 + x^2 + \\frac{1}{16} = 0", "name": null, "statement": "The cubic equation in x^2 obtained by eliminating y^2 between the two real equations for the imaginary roots of z^4 - z + 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the root z" } ], "sympy": "Eq(-4*x**6 + x**2 + Rational(1,16), 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/imaginary-root", "concept/polynomial" ] }, { "id": "dickson-theory-of-equations-1922/eq-b55c9de054", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "114", "location": "Determinants; Systems of Linear Equations", "latex": "a_{11} x_1 + a_{12} x_2 + \\dotsb + a_{1n} x_n = k_1", "name": null, "statement": "The first of n linear equations in n unknowns, whose left side is a weighted sum of the unknowns and whose right side is a known term.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a_{ij}", "meaning": "coefficient of the unknown x_j in the i-th equation" }, { "unit": null, "symbol": "x_i", "meaning": "unknown" }, { "unit": null, "symbol": "k_i", "meaning": "known term of the i-th equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/known-term", "concept/linear-equation", "concept/system-of-linear-equations", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-050cc17f0e", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "114", "location": "Determinants; Systems of Linear Equations", "latex": "D = \\begin{vmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ \\Dots{4} \\\\ a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{vmatrix}", "name": null, "statement": "D is defined as the determinant of the coefficients of the n unknowns.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "D", "meaning": "determinant of the coefficients of the n unknowns" }, { "unit": null, "symbol": "a_{ij}", "meaning": "coefficient of the unknown x_j in the i-th equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/determinant", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-c14e78caa6", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "114", "location": "Determinants; Systems of Linear Equations", "latex": "Dx_1 = K_1,\\qquad Dx_2 = K_2,\\qquad \\dotsc,\\qquad Dx_n = K_n", "name": "Cramer's rule", "statement": "The product of D with each unknown equals the determinant K_i obtained from D by replacing the i-th column of coefficients with the known terms, so each unknown is K_i divided by D when D is nonzero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "D", "meaning": "determinant of the coefficients of the unknowns" }, { "unit": null, "symbol": "x_i", "meaning": "unknown" }, { "unit": null, "symbol": "K_i", "meaning": "determinant formed from D by substituting the known terms for the i-th column of coefficients" } ], "sympy": null, "physics": false, "states": [ "theorem/cramer-s-rule" ], "concepts": [ "concept/determinant", "concept/known-term", "concept/system-of-linear-equations", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-91408a1efd", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "114", "location": "Determinants; Systems of Linear Equations", "latex": "K_1 = \\begin{vmatrix} k_1 & a_{12} & \\cdots & a_{1n} \\\\ \\Dots{4} \\\\ k_n & a_{n2} & \\cdots & a_{nn} \\end{vmatrix}", "name": null, "statement": "K_1 is the determinant obtained from D by replacing the first column of coefficients with the known terms.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "K_1", "meaning": "determinant obtained from D by substituting the known terms for the first column" }, { "unit": null, "symbol": "k_i", "meaning": "known term of the i-th equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/known-term", "theorem/cramer-s-rule" ] }, { "id": "dickson-theory-of-equations-1922/eq-c62045401c", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "118", "location": "Determinants; Systems of Linear Equations", "latex": "L_i \\equiv a_{i1} x_1 + a_{i2} x_2 + \\dotsb + a_{in} x_n - k_i", "name": null, "statement": "L_i is the i-th equation with its known term transposed to the left, so that it is zero when the equation holds.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "L_i", "meaning": "i-th linear equation with known term moved to the left side" }, { "unit": null, "symbol": "a_{ij}", "meaning": "coefficient" }, { "unit": null, "symbol": "k_i", "meaning": "known term" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/known-term", "concept/linear-equation", "method/transposing-the-terms" ] }, { "id": "dickson-theory-of-equations-1922/eq-cb38a63ed9", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "118", "location": "Determinants; Systems of Linear Equations", "latex": "K = \\begin{vmatrix} a_{11} & \\cdots & a_{1r} & k_1 \\\\ \\Dots{4} \\\\ a_{r+11} & \\cdots & a_{r+1r} & k_{r+1} \\end{vmatrix}", "name": null, "statement": "K is the determinant formed from the first r+1 rows by putting the known terms in the last column.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "K", "meaning": "determinant of the first r+1 rows with the known terms as last column" }, { "unit": null, "symbol": "k_i", "meaning": "known term" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/known-term", "concept/minor-arc", "theorem/consistency-of-a-linear-system" ] }, { "id": "dickson-theory-of-equations-1922/eq-2a32685556", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "118", "location": "Determinants; Systems of Linear Equations", "latex": "0 = ±K", "name": null, "statement": "Multiplying and adding the first r+1 equations with the minors of the known terms gives the identity 0 = ±K, which forces K to vanish if the equations are consistent.", "kind": "result", "symbols": [ { "unit": null, "symbol": "K", "meaning": "determinant formed from an (r+1)-rowed minor with the known terms as one column" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/expansion", "concept/linear-combination", "theorem/consistency-of-a-linear-system" ] }, { "id": "dickson-theory-of-equations-1922/eq-1e4d086572", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "118", "location": "Determinants; Systems of Linear Equations", "latex": "d_{r+1} = \\begin{vmatrix} a_{11} & \\cdots & a_{1r} \\\\ \\Dots{3} \\\\ a_{r1} & \\cdots & a_{rr} \\end{vmatrix}", "name": null, "statement": "d_{r+1} is the nonvanishing r-rowed minor formed from the first r rows and first r columns of the coefficient determinant.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "d_{r+1}", "meaning": "r-rowed minor of the coefficients in the first r rows and columns" }, { "unit": null, "symbol": "a_{ij}", "meaning": "coefficient" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/minor-arc", "concept/rank-of-a-determinant" ] }, { "id": "dickson-theory-of-equations-1922/eq-ce8cfb1608", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "118", "location": "Determinants; Systems of Linear Equations", "latex": "d_1L_1 - d_2L_2 + \\dotsb + (-1)^rd_{r+1}L_{r+1} = \\mp K = 0", "name": null, "statement": "A signed combination of the first r+1 equations, with minors as coefficients, reduces to zero, so one equation is a linear combination of the others.", "kind": "result", "symbols": [ { "unit": null, "symbol": "d_i", "meaning": "minor of the known term k_i in K" }, { "unit": null, "symbol": "L_i", "meaning": "i-th equation with known term transposed to the left" }, { "unit": null, "symbol": "K", "meaning": "determinant formed with the known terms as one column" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/linear-combination", "concept/linear-equation", "concept/minor-arc", "theorem/consistency-of-a-linear-system" ] }, { "id": "dickson-theory-of-equations-1922/eq-e1eb73394c", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "120", "location": "Determinants; Systems of Linear Equations", "latex": "A = \\begin{pmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ \\Dots{4} \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{pmatrix}", "name": null, "statement": "A is the matrix of the coefficients of the unknowns, arranged as they occur in the equations.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A", "meaning": "matrix of the coefficients of the unknowns" }, { "unit": null, "symbol": "a_{ij}", "meaning": "coefficient of the unknown x_j in the i-th equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/matrix", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-db2ca6317f", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Determinants; Systems of Linear Equations", "latex": "B = \\begin{pmatrix} a_{11} & a_{12} & \\cdots & a_{1n} & k_1\\\\ \\Dots{5}\\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} & k_m \\end{pmatrix}", "name": "augmented matrix", "statement": "B is the matrix of coefficients with the column of known terms annexed.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "B", "meaning": "augmented matrix of coefficients and known terms" }, { "unit": null, "symbol": "k_i", "meaning": "known term of the i-th equation" } ], "sympy": null, "physics": false, "states": [ "concept/augmented-matrix" ], "concepts": [ "concept/coefficient", "concept/known-term", "concept/matrix" ] }, { "id": "dickson-theory-of-equations-1922/eq-ffb0d502b1", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "122", "location": "Determinants; Systems of Linear Equations", "latex": "D = \\begin{vmatrix} a_1 & b_1 & c_1 & d_1 \\\\ a_2 & b_2 & c_2 & d_2 \\\\ a_3 & b_3 & c_3 & d_3 \\\\ a_4 & b_4 & c_4 & d_4 \\end{vmatrix}", "name": null, "statement": "D is a determinant of order 4 whose elements are the letters a_i, b_i, c_i, d_i, used as the example for complementary minors.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "D", "meaning": "determinant of order 4 used as the example" }, { "unit": null, "symbol": "a_i, b_i, c_i, d_i", "meaning": "elements of the determinant in rows i and columns 1 to 4" } ], "sympy": "Eq(D, Matrix([[a1, b1, c1, d1], [a2, b2, c2, d2], [a3, b3, c3, d3], [a4, b4, c4, d4]]).det())", "physics": false, "states": [], "concepts": [ "concept/complementary-minor", "concept/determinant", "concept/element-of-a-determinant" ] }, { "id": "dickson-theory-of-equations-1922/eq-62f1e40ac4", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "122", "location": "Determinants; Systems of Linear Equations", "latex": "M = \\begin{vmatrix} a_1 & b_1 \\\\ a_3 & b_3 \\end{vmatrix}", "name": null, "statement": "M is the two-rowed minor formed from rows 1 and 3 and columns 1 and 2 of D.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "M", "meaning": "two-rowed complementary minor of D" } ], "sympy": "Eq(M, Matrix([[a1, b1], [a3, b3]]).det())", "physics": false, "states": [], "concepts": [ "concept/complementary-minor", "concept/minor", "concept/minor-arc" ] }, { "id": "dickson-theory-of-equations-1922/eq-a512c0f2f4", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "122", "location": "Determinants; Systems of Linear Equations", "latex": "M' = \\begin{vmatrix} c_2 & d_2 \\\\ c_4 & d_4 \\end{vmatrix}", "name": null, "statement": "M' is the two-rowed minor complementary to M, formed from rows 2 and 4 and columns 3 and 4 of D.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "M'", "meaning": "two-rowed minor complementary to M" } ], "sympy": "Eq(Mp, Matrix([[c2, d2], [c4, d4]]).det())", "physics": false, "states": [], "concepts": [ "concept/complementary-minor", "concept/minor" ] }, { "id": "dickson-theory-of-equations-1922/eq-f7988def99", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "124", "location": "Determinants; Systems of Linear Equations", "latex": "\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} · \\begin{vmatrix} e & f \\\\ g & h \\end{vmatrix} = \\begin{vmatrix} ae + bg & af + bh \\\\ ce + dg & cf + dh \\end{vmatrix}", "name": null, "statement": "The product of two determinants of order 2 equals the determinant whose elements are the row-by-column sums of products.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c, d, e, f, g, h", "meaning": "elements of the two second-order determinants" } ], "sympy": "Eq(Matrix([[a, b], [c, d]]).det()*Matrix([[e, f], [g, h]]).det(), Matrix([[a*e + b*g, a*f + b*h], [c*e + d*g, c*f + d*h]]).det())", "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/element-of-a-determinant", "theorem/product-of-determinants" ] }, { "id": "dickson-theory-of-equations-1922/eq-15b4864de1", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "124", "location": "Determinants; Systems of Linear Equations", "latex": "\\begin{vmatrix} \\Neg a_1 & \\Neg b_1 & \\Neg c_1 & 0 & 0 & 0 \\\\ \\Neg a_2 & \\Neg b_2 & \\Neg c_2 & 0 & 0 & 0 \\\\ \\Neg a_3 & \\Neg b_3 & \\Neg c_3 & 0 & 0 & 0 \\\\ -1 & \\Neg 0 & \\Neg 0 & e_1 & f_1 & g_1 \\\\ \\Neg 0 & -1 & \\Neg 0 & e_2 & f_2 & g_2 \\\\ \\Neg 0 & \\Neg 0 & -1 & e_3 & f_3 & g_3 \\end{vmatrix} = \\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix} · \\begin{vmatrix} e_1 & f_1 & g_1 \\\\ e_2 & f_2 & g_2 \\\\ e_3 & f_3 & g_3 \\end{vmatrix}", "name": null, "statement": "The product of two third-order determinants equals a determinant of order 6, obtained by a Laplace development with r = 3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_i, b_i, c_i", "meaning": "elements of the first third-order determinant" }, { "unit": null, "symbol": "e_i, f_i, g_i", "meaning": "elements of the second third-order determinant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "method/laplace-s-development", "theorem/product-of-determinants" ] }, { "id": "dickson-theory-of-equations-1922/eq-78c942b9d6", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Determinants; Systems of Linear Equations", "latex": "x_1 = 0, \\dotsc, x_n = 0", "name": "trivial solution", "statement": "Every unknown equal to zero satisfies any set of homogeneous linear equations, so this is always a solution.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_i", "meaning": "unknown" } ], "sympy": null, "physics": false, "states": [ "concept/homogeneous-linear-equations" ], "concepts": [ "concept/solution", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-a354ca37fe", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "111", "location": "Determinants; Systems of Linear Equations", "latex": "D = \\sum_{j=1}^n (-1)^{j+k} e_{jk} E_{jk}", "name": null, "statement": "Any determinant of order n equals the signed sum of the elements of its kth column times their minors (expansion according to any column).", "kind": "result", "symbols": [ { "unit": null, "symbol": "D", "meaning": "determinant of order n" }, { "unit": null, "symbol": "e_{jk}", "meaning": "element in row j, column k of D" }, { "unit": null, "symbol": "E_{jk}", "meaning": "minor of e_{jk} in D" }, { "unit": null, "symbol": "k", "meaning": "index of the column expanded along" }, { "unit": null, "symbol": "n", "meaning": "order of the determinant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/element-of-a-determinant", "concept/expansion", "concept/minor-arc", "concept/rule" ] }, { "id": "dickson-theory-of-equations-1922/eq-df8a4e28cd", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "101", "location": "Determinants; Systems of Linear Equations", "latex": "a_1 b_2 - a_2 b_1", "name": null, "statement": "The determinant of the second order is defined as the common multiplier a_1 b_2 - a_2 b_1 of x and y in the two-equation system.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_1", "meaning": "coefficient of x in the first equation" }, { "unit": null, "symbol": "b_1", "meaning": "coefficient of y in the first equation" }, { "unit": null, "symbol": "a_2", "meaning": "coefficient of x in the second equation" }, { "unit": null, "symbol": "b_2", "meaning": "coefficient of y in the second equation" } ], "sympy": "Eq(D, a1*b2 - a2*b1)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/determinant", "concept/method-solving-simultaneous-equations-by-determinants", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-e9aa4680d4", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "102", "location": "Determinants; Systems of Linear Equations", "latex": "x = \\frac{k_1 b_2 - k_2 b_1}{D}", "name": null, "statement": "When D is not zero, the unknown x equals the determinant with the known terms substituted in the first column, divided by D.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "first unknown" }, { "unit": null, "symbol": "k_1", "meaning": "known term of the first equation" }, { "unit": null, "symbol": "k_2", "meaning": "known term of the second equation" }, { "unit": null, "symbol": "b_1", "meaning": "coefficient of y in the first equation" }, { "unit": null, "symbol": "b_2", "meaning": "coefficient of y in the second equation" }, { "unit": null, "symbol": "D", "meaning": "determinant of the coefficients of the unknowns" } ], "sympy": "Eq(x, (k1*b2 - k2*b1)/D)", "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/known-term", "concept/method-solving-simultaneous-equations-by-determinants", "concept/theorem-cramer-s-rule", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-b971437ccd", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "102", "location": "Determinants; Systems of Linear Equations", "latex": "y = \\frac{a_1 k_2 - a_2 k_1}{D}", "name": null, "statement": "When D is not zero, the unknown y equals the determinant with the known terms substituted in the second column, divided by D.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "second unknown" }, { "unit": null, "symbol": "a_1", "meaning": "coefficient of x in the first equation" }, { "unit": null, "symbol": "a_2", "meaning": "coefficient of x in the second equation" }, { "unit": null, "symbol": "k_1", "meaning": "known term of the first equation" }, { "unit": null, "symbol": "k_2", "meaning": "known term of the second equation" }, { "unit": null, "symbol": "D", "meaning": "determinant of the coefficients of the unknowns" } ], "sympy": "Eq(y, (a1*k2 - a2*k1)/D)", "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/known-term", "concept/method-solving-simultaneous-equations-by-determinants", "concept/theorem-cramer-s-rule", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/eq-ce9c5ced09", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "102", "location": "Determinants; Systems of Linear Equations", "latex": "a_1 b_2 c_3 - a_1 b_3 c_2 + a_2 b_3 c_1 - a_2 b_1 c_3 + a_3 b_1 c_2 - a_3 b_2 c_1", "name": null, "statement": "The determinant of the third order is defined as the signed sum of the six products taking one element from each row and column, with signs fixed by the parity of the subscript arrangement.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_1", "meaning": "element in row 1, column 1 of the determinant" }, { "unit": null, "symbol": "b_1", "meaning": "element in row 1, column 2 of the determinant" }, { "unit": null, "symbol": "c_1", "meaning": "element in row 1, column 3 of the determinant" }, { "unit": null, "symbol": "a_2", "meaning": "element in row 2, column 1 of the determinant" }, { "unit": null, "symbol": "b_2", "meaning": "element in row 2, column 2 of the determinant" }, { "unit": null, "symbol": "c_2", "meaning": "element in row 2, column 3 of the determinant" }, { "unit": null, "symbol": "a_3", "meaning": "element in row 3, column 1 of the determinant" }, { "unit": null, "symbol": "b_3", "meaning": "element in row 3, column 2 of the determinant" }, { "unit": null, "symbol": "c_3", "meaning": "element in row 3, column 3 of the determinant" } ], "sympy": "Eq(D, a1*b2*c3 - a1*b3*c2 + a2*b3*c1 - a2*b1*c3 + a3*b1*c2 - a3*b2*c1)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/determinant", "concept/diagonal-term-of-a-determinant", "concept/element-of-a-determinant", "concept/permutation", "concept/zodiacal-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-daab0d8ccf", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "105", "location": "Determinants; Systems of Linear Equations", "latex": "\\sum_{(24)} ± a_q b_r c_s d_t", "name": null, "statement": "A determinant of order 4 is the sum over all 24 arrangements q, r, s, t of 1, 2, 3, 4 of the products a_q b_r c_s d_t, each with the sign + or - according to the parity of the arrangement.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "first column element of the order-4 determinant" }, { "unit": null, "symbol": "b", "meaning": "second column element of the order-4 determinant" }, { "unit": null, "symbol": "c", "meaning": "third column element of the order-4 determinant" }, { "unit": null, "symbol": "d", "meaning": "fourth column element of the order-4 determinant" }, { "unit": null, "symbol": "q", "meaning": "row index for the first column in an arrangement of 1, 2, 3, 4" }, { "unit": null, "symbol": "r", "meaning": "row index for the second column in an arrangement of 1, 2, 3, 4" }, { "unit": null, "symbol": "s", "meaning": "row index for the third column in an arrangement of 1, 2, 3, 4" }, { "unit": null, "symbol": "t", "meaning": "row index for the fourth column in an arrangement of 1, 2, 3, 4" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/arrangement", "concept/determinant", "concept/permutation", "concept/sum", "concept/zodiacal-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-497c76589d", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "106", "location": "Determinants; Systems of Linear Equations", "latex": "(-1)^i e_{{i_1}1} e_{{i_2}2} \\dotsm e_{{i_n}n}", "name": null, "statement": "A term of a determinant of order n is a product with one element from each row and column, with sign (-1) to the power i, where i is the number of interchanges that produce the arrangement i_1, ..., i_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "i", "meaning": "number of interchanges needed to derive the arrangement i_1, ..., i_n from 1, ..., n" }, { "unit": null, "symbol": "i_1, ..., i_n", "meaning": "an arrangement of 1, 2, ..., n" }, { "unit": null, "symbol": "e_{jk}", "meaning": "element in row j, column k of the determinant" }, { "unit": null, "symbol": "n", "meaning": "order of the determinant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/arrangement", "concept/determinant", "concept/element-of-a-determinant", "concept/permutation", "concept/product", "concept/zodiacal-sign" ] }, { "id": "dickson-theory-of-equations-1922/eq-dbf1e7bd6c", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "110", "location": "Determinants; Systems of Linear Equations", "latex": "D = e_{11}E_{11} - e_{21}E_{21} + e_{31}E_{31} - \\dotsb + (-1)^{n-1} e_{n1}E_{n1}", "name": null, "statement": "Any determinant of order n equals the alternating sum of its first-column elements times their minors (expansion according to the first column).", "kind": "result", "symbols": [ { "unit": null, "symbol": "D", "meaning": "determinant of order n" }, { "unit": null, "symbol": "e_{ij}", "meaning": "element in row i, column j of D" }, { "unit": null, "symbol": "E_{ij}", "meaning": "minor of e_{ij} in D" }, { "unit": null, "symbol": "n", "meaning": "order of the determinant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/element-of-a-determinant", "concept/expansion", "concept/minor-arc", "concept/rule" ] }, { "id": "dickson-theory-of-equations-1922/eq-5e61aafa7e", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "109", "location": "Determinants; Systems of Linear Equations", "latex": "D = -a_2A_2 + b_2B_2 - c_2C_2", "name": null, "statement": "A third-order determinant equals the expansion along its second row, with alternating signs applied to each element times its minor.", "kind": "result", "symbols": [ { "unit": null, "symbol": "D", "meaning": "determinant of the third order" }, { "unit": null, "symbol": "a_2", "meaning": "element in row 2, column 1" }, { "unit": null, "symbol": "b_2", "meaning": "element in row 2, column 2" }, { "unit": null, "symbol": "c_2", "meaning": "element in row 2, column 3" }, { "unit": null, "symbol": "A_2", "meaning": "minor of a_2" }, { "unit": null, "symbol": "B_2", "meaning": "minor of b_2" }, { "unit": null, "symbol": "C_2", "meaning": "minor of c_2" } ], "sympy": "Eq(D, -a2*A2 + b2*B2 - c2*C2)", "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/element-of-a-determinant", "concept/expansion", "concept/minor-arc" ] }, { "id": "dickson-theory-of-equations-1922/eq-f135ad4b00", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "109", "location": "Determinants; Systems of Linear Equations", "latex": "D = -b_1B_1 + b_2B_2 - b_3B_3", "name": null, "statement": "A third-order determinant equals the expansion along its second column, with alternating signs applied to each element times its minor.", "kind": "result", "symbols": [ { "unit": null, "symbol": "D", "meaning": "determinant of the third order" }, { "unit": null, "symbol": "b_1", "meaning": "element in row 1, column 2" }, { "unit": null, "symbol": "b_2", "meaning": "element in row 2, column 2" }, { "unit": null, "symbol": "b_3", "meaning": "element in row 3, column 2" }, { "unit": null, "symbol": "B_1", "meaning": "minor of b_1" }, { "unit": null, "symbol": "B_2", "meaning": "minor of b_2" }, { "unit": null, "symbol": "B_3", "meaning": "minor of b_3" } ], "sympy": "Eq(D, -b1*B1 + b2*B2 - b3*B3)", "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/element-of-a-determinant", "concept/expansion", "concept/minor-arc" ] }, { "id": "dickson-theory-of-equations-1922/eq-abfb6d9f72", "chapter": "dickson-theory-of-equations-1922/ch-viii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "105", "location": "Determinants; Systems of Linear Equations", "latex": "(-1)^m P \\equiv (-1)^t P", "name": null, "statement": "If one arrangement is reached from 1, 2, ..., n by m interchanges and also by t interchanges, then m and t are both even or both odd.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of interchanges in one derivation of an arrangement" }, { "unit": null, "symbol": "t", "meaning": "number of interchanges in another derivation of the same arrangement" }, { "unit": null, "symbol": "P", "meaning": "product of all differences x_i - x_j (i \\rho^n(1-p) \\geqq P", "name": null, "statement": "Shows that |f(z)| exceeds any preassigned P once rho is large enough.", "kind": "result", "symbols": [ { "unit": null, "symbol": "|f(z)|", "meaning": "absolute value of f(z)" }, { "unit": null, "symbol": "rho", "meaning": "absolute value of z" }, { "unit": null, "symbol": "p", "meaning": "assigned positive number less than 1" }, { "unit": null, "symbol": "P", "meaning": "assigned positive number" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/inequality", "concept/infinity", "concept/polynomial", "concept/sufficiently-large-values" ] }, { "id": "dickson-theory-of-equations-1922/eq-63021b5876", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "156", "location": "Appendix", "latex": "\\rho \\geqq \\sqrt[n]{\\frac{P}{1-p}} \\equiv R", "name": null, "statement": "Defines the radius R so that |z| at least R guarantees |f(z)| greater than P.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "absolute value of z" }, { "unit": null, "symbol": "R", "meaning": "radius defined in Lemma 2" }, { "unit": null, "symbol": "P", "meaning": "assigned positive number" }, { "unit": null, "symbol": "p", "meaning": "assigned positive number less than 1" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/exponent", "concept/root", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-121cb70165", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "156", "location": "Appendix", "latex": "f(a+h) = f(a) + f'(a)h + \\dotsb + f^{(r)}(a)·\\frac{h^r}{r!} + \\dotsb + f^{(n)}(a)·\\frac{h^n}{n!}", "name": "Taylor's formula", "statement": "Expands f(a+h) in powers of h, with derivatives of f at a as coefficients.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(a+h)", "meaning": "value of the polynomial at a + h" }, { "unit": null, "symbol": "a", "meaning": "complex number about which f is expanded" }, { "unit": null, "symbol": "h", "meaning": "complex increment" }, { "unit": null, "symbol": "f^{(r)}(a)", "meaning": "r-th derivative of f at a" }, { "unit": null, "symbol": "r!", "meaning": "r factorial" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" } ], "sympy": null, "physics": false, "states": [ "theorem/taylor-s-theorem" ], "concepts": [ "concept/derivative", "concept/factorial", "concept/polynomial", "concept/power" ] }, { "id": "dickson-theory-of-equations-1922/eq-55fdbd4dce", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "156", "location": "Appendix", "latex": "g(h) \\equiv 1 + bh^r + ch^{r+1} + \\dotsb + lh^n", "name": null, "statement": "Defines the simplified function g(h) as the normalized expansion of f(a+h) divided by f(a).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "g(h)", "meaning": "normalized polynomial in h" }, { "unit": null, "symbol": "h", "meaning": "complex increment" }, { "unit": null, "symbol": "b, c, l", "meaning": "complex coefficients in the simplified expansion" }, { "unit": null, "symbol": "r, n", "meaning": "exponents with r less than n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/exponent", "concept/function", "concept/power" ] }, { "id": "dickson-theory-of-equations-1922/eq-aae1ab37e0", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "156", "location": "Appendix", "latex": "h = \\rho(\\cos \\theta + i \\sin \\theta)", "name": null, "statement": "Writes the complex increment h in trigonometric form with modulus rho and argument theta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "h", "meaning": "complex increment" }, { "unit": null, "symbol": "rho", "meaning": "modulus of h" }, { "unit": "degree of angle", "symbol": "theta", "meaning": "argument of h" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(h, rho*(cos(theta) + I*sin(theta)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "concept/trigonometric-form-of-a-complex-number", "quantity/angle", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-ab0d37c77b", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "156", "location": "Appendix", "latex": "b = |b|(\\cos \\beta + i \\sin \\beta)", "name": null, "statement": "Writes the complex coefficient b in trigonometric form with modulus |b| and argument beta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "b", "meaning": "complex coefficient in g(h)" }, { "unit": null, "symbol": "|b|", "meaning": "modulus of b" }, { "unit": "degree of angle", "symbol": "beta", "meaning": "argument of b" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(b, Abs(b)*(cos(beta) + I*sin(beta)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "concept/trigonometric-form-of-a-complex-number", "quantity/angle", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-d198694967", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "bh^r = |b| \\rho^r \\bigl\\{\\cos(\\beta+r\\theta) + i\\sin (\\beta+r\\theta)\\bigr\\}", "name": null, "statement": "The product b h^r in trigonometric form, with argument beta plus r theta, by de Moivre's rule.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "b h^r", "meaning": "complex term of g(h)" }, { "unit": null, "symbol": "|b|", "meaning": "modulus of b" }, { "unit": null, "symbol": "rho", "meaning": "modulus of h" }, { "unit": null, "symbol": "r", "meaning": "exponent of the leading term" }, { "unit": "degree of angle", "symbol": "beta", "meaning": "argument of b" }, { "unit": "degree of angle", "symbol": "theta", "meaning": "argument of h" } ], "sympy": "Eq(b*h**r, Abs(b)*rho**r*(cos(beta + r*theta) + I*sin(beta + r*theta)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/power", "concept/sine", "concept/trigonometric-form-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-7b5eabbee8", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "g(h) = (1 - |b|\\rho^r) + h^r(ch + \\dotsb + lh^{n-r})", "name": null, "statement": "Rewrites g(h) so that its size is controlled by the first term 1 minus |b| rho^r.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "g(h)", "meaning": "normalized polynomial in h" }, { "unit": null, "symbol": "|b|", "meaning": "modulus of b" }, { "unit": null, "symbol": "rho", "meaning": "modulus of h" }, { "unit": null, "symbol": "r", "meaning": "exponent of the leading term" }, { "unit": null, "symbol": "c, l", "meaning": "complex coefficients in g(h)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/exponent", "concept/function", "concept/power" ] }, { "id": "dickson-theory-of-equations-1922/eq-ea00ba0510", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "|ch + \\dotsb + lh^{n-r}| < |b|", "name": null, "statement": "Chooses rho small enough that the remaining terms of g(h) are smaller in absolute value than |b|.", "kind": "result", "symbols": [ { "unit": null, "symbol": "|b|", "meaning": "modulus of b" }, { "unit": null, "symbol": "h", "meaning": "complex increment" }, { "unit": null, "symbol": "c, l", "meaning": "complex coefficients in g(h)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/coefficient", "concept/inequality" ] }, { "id": "dickson-theory-of-equations-1922/eq-3cc9f23397", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "|b| \\rho^r < 1", "name": null, "statement": "Chooses rho small enough that the product |b| rho^r is less than 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "|b|", "meaning": "modulus of b" }, { "unit": null, "symbol": "rho", "meaning": "modulus of h" }, { "unit": null, "symbol": "r", "meaning": "exponent of the leading term" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/inequality", "concept/power" ] }, { "id": "dickson-theory-of-equations-1922/eq-b80fa895a6", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "|g(h)| < (1 - |b|\\rho^r) + \\rho^r|b|", "name": null, "statement": "Bounds |g(h)| by a quantity that is less than 1, so |f(a+h)/f(a)| is less than 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "|g(h)|", "meaning": "absolute value of g(h)" }, { "unit": null, "symbol": "|b|", "meaning": "modulus of b" }, { "unit": null, "symbol": "rho", "meaning": "modulus of h" }, { "unit": null, "symbol": "r", "meaning": "exponent of the leading term" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/inequality", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-448e97d4c4", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "G(x,y) = \\phi^2(x, y) + \\psi^2(x, y)", "name": null, "statement": "Defines G as the sum of the squares of the real and imaginary parts of f, so that G is the square of |f(z)|.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "G(x,y)", "meaning": "sum of squares of phi and psi" }, { "unit": null, "symbol": "phi, psi", "meaning": "real polynomials in x and y" }, { "unit": null, "symbol": "x, y", "meaning": "real coordinates of z = x + iy" } ], "sympy": "Eq(G(x, y), phi(x, y)**2 + psi(x, y)**2)", "physics": false, "states": [], "concepts": [ "concept/function-of-two-variables", "concept/polynomial", "concept/real-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-47be3a5cd5", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "157", "location": "Appendix", "latex": "x^2 + y^2 = R^2", "name": null, "statement": "The circle C of radius R in the plane of the real and imaginary parts of z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "real coordinates of z = x + iy" }, { "unit": null, "symbol": "R", "meaning": "radius of the circle C" } ], "sympy": "Eq(x**2 + y**2, R**2)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/coordinate-geometry", "concept/square" ] }, { "id": "dickson-theory-of-equations-1922/eq-cbdb6ee0b7", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "158", "location": "Appendix", "latex": "G(x_1,y_1) \\leqq G(x,y)", "name": null, "statement": "The value G(x_1, y_1) is a minimum of G over the closed disc x^2 + y^2 at most R^2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "G(x_1,y_1)", "meaning": "minimum value of G over the disc" }, { "unit": null, "symbol": "x_1, y_1", "meaning": "real pair at which G is a minimum" }, { "unit": null, "symbol": "x, y", "meaning": "real coordinates of any point in the disc" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/function-of-two-variables", "concept/inequality", "concept/minimum" ] }, { "id": "dickson-theory-of-equations-1922/eq-b365c1e9da", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "158", "location": "Appendix", "latex": "|f(z)|^2 = G(x,y)", "name": null, "statement": "The square of the absolute value of f(z) equals G(x, y).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "|f(z)|", "meaning": "absolute value of f(z)" }, { "unit": null, "symbol": "G(x,y)", "meaning": "sum of squares of phi and psi" } ], "sympy": "Eq(Abs(f(z))**2, G(x, y))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/function-of-two-variables", "quantity/modulus-of-a-complex-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-802dd70860", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "158", "location": "Appendix", "latex": "|f(z_1)|\\leqq |f(z)|", "name": null, "statement": "The absolute value of f is smallest at z_1 among all z on or within the circle C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z_1", "meaning": "complex number x_1 + i y_1 at which G is minimal" }, { "unit": null, "symbol": "z", "meaning": "any complex number in or on the circle C" }, { "unit": null, "symbol": "f", "meaning": "polynomial in z" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/complex-number", "concept/function", "concept/minimum" ] }, { "id": "dickson-theory-of-equations-1922/eq-c13a38941d", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "158", "location": "Appendix", "latex": "|f(z_1)|\\leqq |f(z')| < P", "name": null, "statement": "The minimum value of |f| on the circle C is at most |f(z')| and therefore less than P.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z_1", "meaning": "complex number x_1 + i y_1 at which G is minimal" }, { "unit": null, "symbol": "z'", "meaning": "any complex number with f(z') nonzero" }, { "unit": null, "symbol": "P", "meaning": "positive number exceeding |f(z')|" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/complex-number", "concept/inequality", "concept/minimum" ] }, { "id": "dickson-theory-of-equations-1922/eq-7ae83c12ce", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "158", "location": "Appendix", "latex": "|f(z)| < |f(z_1)|", "name": null, "statement": "If f(z_1) were nonzero, some z would give a smaller |f(z)| than |f(z_1)|, which contradicts the minimum.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number whose |f(z)| is smaller than at z_1" }, { "unit": null, "symbol": "z_1", "meaning": "complex number x_1 + i y_1 at which G is minimal" }, { "unit": null, "symbol": "f", "meaning": "polynomial in z" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/inequality", "concept/minimum", "concept/root" ] }, { "id": "dickson-theory-of-equations-1922/eq-a0687933dd", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "156", "location": "Appendix", "latex": "f^{(n)}(a) = n!", "name": null, "statement": "The n-th derivative of f is the constant n factorial, because the leading coefficient of f is 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f^{(n)}(a)", "meaning": "n-th derivative of f at a" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/factorial", "concept/polynomial", "unit/degree-of-angle" ] }, { "id": "dickson-theory-of-equations-1922/eq-188881ec7f", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "155", "location": "Appendix", "latex": "z = x+iy", "name": null, "statement": "Writes the complex variable z in terms of its real part x and imaginary part y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" }, { "unit": null, "symbol": "x, y", "meaning": "real numbers, the real and imaginary parts of z" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(z, x + I*y)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/real-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-538016afb8", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "155", "location": "Appendix", "latex": "a_1 = c_1 + id_1", "name": null, "statement": "Writes the complex coefficient a_1 with real part c_1 and imaginary part d_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_1", "meaning": "complex coefficient of the equation" }, { "unit": null, "symbol": "c_1, d_1", "meaning": "real parts of a_1" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(a_1, c_1 + I*d_1)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/complex-number", "concept/real-number" ] }, { "id": "dickson-theory-of-equations-1922/eq-ca1ad27dcb", "chapter": "dickson-theory-of-equations-1922/ch-appendix", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "156", "location": "Appendix", "latex": "\\rho\\equiv |z|", "name": null, "statement": "Defines rho as the absolute value of z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "absolute value of z" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": "Eq(rho, Abs(z))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "quantity/modulus-of-a-complex-number" ] } ], "exercise_sets": [ { "id": "dickson-theory-of-equations-1922/ex-page2", "set": "Page2", "page": "2", "chapter": "dickson-theory-of-equations-1922/ch-i", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6", "set": "Page6", "page": "6", "chapter": "dickson-theory-of-equations-1922/ch-i", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page9", "set": "Page9", "page": "9", "chapter": "dickson-theory-of-equations-1922/ch-i", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10", "set": "Page10", "page": "10", "chapter": "dickson-theory-of-equations-1922/ch-i", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13", "set": "Page13", "page": "13", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [ "concept/compound-interest", "concept/divisibility", "concept/factor", "concept/geometrical-progression", "concept/polynomial", "theorem/factor-theorem", "theorem/remainder-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page15", "set": "Page15", "page": "15", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [ "concept/dividend", "concept/divisor", "concept/polynomial", "concept/quadratic-equation", "concept/quotient", "concept/remainder", "concept/root-of-an-equation", "method/synthetic-division", "theorem/factor-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page17", "set": "Page17", "page": "17", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [ "concept/discriminant", "concept/equation", "concept/identity", "concept/multiple-root", "concept/quadratic-equation", "concept/root-of-an-equation", "method/equating-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/ex-page19", "set": "Page19", "page": "19", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [ "concept/arithmetical-progression", "concept/coefficient", "concept/common-difference", "concept/common-ratio", "concept/geometrical-progression", "concept/multiple-root", "concept/root-of-an-equation", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "dickson-theory-of-equations-1922/ex-page20", "set": "Page20", "page": "20", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [ "concept/factored-form", "concept/imaginary-root", "concept/irrational-number", "concept/polynomial", "concept/rational-number", "concept/rational-root", "concept/real-root", "concept/root-of-an-equation", "theorem/imaginary-roots-occur-in-pairs" ] }, { "id": "dickson-theory-of-equations-1922/ex-page23", "set": "Page23", "page": "23", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [ "concept/absolute-value", "concept/complex-number", "concept/lower-limit-to-the-roots", "concept/negative-number", "concept/real-root", "concept/upper-limit-to-the-roots", "theorem/upper-limit-theorem-by-coefficient-ratios", "theorem/upper-limit-theorem-by-greatest-negative-coefficient" ] }, { "id": "dickson-theory-of-equations-1922/ex-page25", "set": "Page25", "page": "25", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page26", "set": "Page26", "page": "26", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page27", "set": "Page27", "page": "27", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28", "set": "Page28", "page": "28", "chapter": "dickson-theory-of-equations-1922/ch-ii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page30", "set": "Page30", "page": "30", "chapter": "dickson-theory-of-equations-1922/ch-iii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40", "set": "Page40", "page": "39", "chapter": "dickson-theory-of-equations-1922/ch-iii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44", "set": "Page44", "page": "44", "chapter": "dickson-theory-of-equations-1922/ch-iii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page46", "set": "Page46", "page": "46", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [ "concept/cubic-equation", "concept/reduced-cubic-equation", "method/extracting-the-cube-root", "method/substitution", "theorem/cardan-s-formulas" ] }, { "id": "dickson-theory-of-equations-1922/ex-page48", "set": "Page48", "page": "48", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [ "concept/cubic-equation", "concept/discriminant", "concept/multiple-root", "concept/real-root", "theorem/number-of-real-roots-of-a-cubic" ] }, { "id": "dickson-theory-of-equations-1922/ex-page49", "set": "Page49", "page": "49", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [ "concept/cubic-equation", "concept/irreducible-case", "concept/reduced-cubic-equation", "theorem/cardan-s-formulas" ] }, { "id": "dickson-theory-of-equations-1922/ex-page51", "set": "Page51", "page": "51", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [ "concept/quartic-equation", "concept/resolvent-cubic", "method/completing-the-square", "method/ferrari-s-solution-of-the-quartic-equation" ] }, { "id": "dickson-theory-of-equations-1922/ex-page52", "set": "Page52", "page": "52", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page53", "set": "Page53", "page": "53", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54", "set": "Page54", "page": "54", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [ "concept/cubic-equation", "concept/discriminant", "concept/imaginary-root", "concept/multiple-root", "concept/quartic-equation", "concept/real-root" ] }, { "id": "dickson-theory-of-equations-1922/ex-page55", "set": "Page55", "page": "55", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/graph-of-a-function", "concept/imaginary-root", "concept/quadratic-equation", "concept/real-root", "method/graphical-solution-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/ex-page59", "set": "Page59", "page": "59", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/bend-point", "concept/derivative", "concept/graph-of-a-function", "concept/higher-order-derivative", "concept/real-root", "concept/root-of-an-equation", "concept/slope-of-a-curve" ] }, { "id": "dickson-theory-of-equations-1922/ex-page62", "set": "Page62", "page": "62", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/derivative", "concept/highest-common-factor", "concept/multiple-root" ] }, { "id": "dickson-theory-of-equations-1922/ex-page64", "set": "Page64", "page": "64", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/bend-point", "concept/inflection-point", "concept/inflection-tangent", "concept/real-root", "concept/reduced-cubic-equation" ] }, { "id": "dickson-theory-of-equations-1922/ex-page66", "set": "Page66", "page": "66", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/bend-point", "concept/cubic-equation", "concept/graph-of-a-function", "concept/real-root" ] }, { "id": "dickson-theory-of-equations-1922/ex-page68", "set": "Page68", "page": "68", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/conjugate-complex-numbers", "concept/imaginary-root", "concept/multiple-root", "concept/real-root", "theorem/intermediate-value-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page69", "set": "Page69", "page": "69", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/infinity", "concept/polynomial", "concept/real-root", "concept/sign-of-a-polynomial-at-infinity" ] }, { "id": "dickson-theory-of-equations-1922/ex-page70", "set": "Page70", "page": "70", "chapter": "dickson-theory-of-equations-1922/ch-v", "practices": [ "concept/bend-point", "concept/derivative", "concept/imaginary-root", "concept/real-root", "theorem/rolle-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page74", "set": "Page74", "page": "74", "chapter": "dickson-theory-of-equations-1922/ch-vi", "practices": [ "concept/imaginary-root", "concept/real-root", "concept/variation-of-sign", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/ex-page78", "set": "Page78", "page": "78", "chapter": "dickson-theory-of-equations-1922/ch-vi", "practices": [ "concept/real-root", "method/isolation-of-a-root", "theorem/sturm-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page79", "set": "Page79", "page": "79", "chapter": "dickson-theory-of-equations-1922/ch-vi", "practices": [ "concept/cubic-equation", "concept/discriminant", "concept/multiple-root", "concept/real-root", "theorem/sturm-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page81", "set": "Page81", "page": "81", "chapter": "dickson-theory-of-equations-1922/ch-vi", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page83", "set": "Page83", "page": "83", "chapter": "dickson-theory-of-equations-1922/ch-vi", "practices": [ "concept/multiple-root", "concept/real-root", "theorem/sturm-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page85", "set": "Page85", "page": "85", "chapter": "dickson-theory-of-equations-1922/ch-vi", "practices": [ "concept/higher-order-derivative", "concept/real-root", "concept/root-of-an-equation", "theorem/budan-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page89", "set": "Page89", "page": "89", "chapter": "dickson-theory-of-equations-1922/ch-vii", "practices": [ "concept/compound-interest", "concept/cosine", "concept/real-root", "concept/right-circular-cylinder", "concept/root-of-an-equation", "concept/specific-gravity", "method/horner-s-method", "method/isolation-of-a-root", "quantity/rate-of-interest" ] }, { "id": "dickson-theory-of-equations-1922/ex-page94", "set": "Page94", "page": "94", "chapter": "dickson-theory-of-equations-1922/ch-vii", "practices": [ "concept/bend-point", "concept/derivative", "concept/higher-order-derivative", "concept/inflection-point", "method/newton-s-method", "theorem/descartes-rule-of-signs" ] }, { "id": "dickson-theory-of-equations-1922/ex-page96", "set": "Page96", "page": "96", "chapter": "dickson-theory-of-equations-1922/ch-vii", "practices": [ "concept/root-of-an-equation", "concept/transformed-equation", "method/newton-s-method", "method/synthetic-division", "theorem/taylor-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page98", "set": "Page98", "page": "98", "chapter": "dickson-theory-of-equations-1922/ch-vii", "practices": [ "concept/circular-segment", "concept/common-logarithm", "concept/logarithm", "concept/sine", "method/interpolating-in-logarithmic-tables", "method/newton-s-method", "method/regula-falsi", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/ex-page99", "set": "Page99", "page": "99", "chapter": "dickson-theory-of-equations-1922/ch-vii", "practices": [ "concept/cubic-equation", "concept/imaginary-root", "concept/quartic-equation", "concept/root-of-an-equation", "theorem/taylor-s-theorem" ] }, { "id": "dickson-theory-of-equations-1922/ex-page100", "set": "Page100", "page": "100", "chapter": "dickson-theory-of-equations-1922/ch-vii", "practices": [ "concept/chord", "concept/circular-segment", "concept/compound-interest", "concept/cotangent", "concept/logarithm", "concept/root-of-an-equation", "concept/simple-interest", "concept/sine", "concept/specific-gravity", "concept/tangent-function", "method/newton-s-method", "quantity/arc-of-a-circle", "unit/radian" ] }, { "id": "dickson-theory-of-equations-1922/ex-page102", "set": "Page102", "page": "102", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [ "concept/determinant", "concept/simultaneous-equations", "method/solving-simultaneous-equations-by-determinants" ] }, { "id": "dickson-theory-of-equations-1922/ex-page104", "set": "Page104", "page": "104", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page106", "set": "Page106", "page": "106", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page108", "set": "Page108", "page": "108", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page112", "set": "Page112", "page": "112", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page113", "set": "Page113", "page": "113", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115", "set": "Page115", "page": "115", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [ "concept/determinant", "concept/linear-equation", "concept/unknown", "method/solving-simultaneous-equations-by-determinants", "theorem/cramer-s-rule" ] }, { "id": "dickson-theory-of-equations-1922/ex-page119", "set": "Page119", "page": "119", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [ "concept/linear-equation", "concept/rank-of-a-determinant", "concept/unknown", "theorem/consistency-of-a-linear-system" ] }, { "id": "dickson-theory-of-equations-1922/ex-page120", "set": "Page120", "page": "120", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [ "concept/homogeneous-linear-equations", "concept/linear-equation", "concept/rank-of-a-determinant", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/ex-page121", "set": "Page121", "page": "121", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [ "concept/augmented-matrix", "concept/homogeneous-linear-equations", "concept/linear-equation", "concept/matrix", "concept/rank-of-a-determinant", "theorem/consistency-of-a-linear-system" ] }, { "id": "dickson-theory-of-equations-1922/ex-page124", "set": "Page124", "page": "124", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page125", "set": "Page125", "page": "125", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126", "set": "Page126", "page": "126", "chapter": "dickson-theory-of-equations-1922/ch-viii", "practices": [ "concept/circle", "concept/cube-root-of-unity", "concept/cubic-equation", "concept/determinant", "concept/factor", "concept/homogeneous-linear-equations", "concept/linear-equation", "concept/unknown" ] }, { "id": "dickson-theory-of-equations-1922/ex-page129", "set": "Page129", "page": "129", "chapter": "dickson-theory-of-equations-1922/ch-ix", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133", "set": "Page133", "page": "133", "chapter": "dickson-theory-of-equations-1922/ch-ix", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136", "set": "Page136", "page": "136", "chapter": "dickson-theory-of-equations-1922/ch-ix", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page140", "set": "Page140", "page": "140", "chapter": "dickson-theory-of-equations-1922/ch-ix", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141", "set": "Page141", "page": "141", "chapter": "dickson-theory-of-equations-1922/ch-ix", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142", "set": "Page142", "page": "142", "chapter": "dickson-theory-of-equations-1922/ch-ix", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page144", "set": "Page144", "page": "144", "chapter": "dickson-theory-of-equations-1922/ch-x", "practices": [ "concept/degree", "concept/elementary-symmetric-function", "concept/polynomial", "concept/resultant", "concept/root-of-an-equation" ] }, { "id": "dickson-theory-of-equations-1922/ex-page147", "set": "Page147", "page": "147", "chapter": "dickson-theory-of-equations-1922/ch-x", "practices": [ "concept/determinant", "concept/resultant", "method/laplace-s-development", "method/sylvester-s-dialytic-method" ] }, { "id": "dickson-theory-of-equations-1922/ex-page150", "set": "Page150", "page": "150", "chapter": "dickson-theory-of-equations-1922/ch-x", "practices": [ "concept/determinant", "concept/resultant", "method/b-zout-s-method-of-elimination" ] }, { "id": "dickson-theory-of-equations-1922/ex-page152", "set": "Page152", "page": "152", "chapter": "dickson-theory-of-equations-1922/ch-x", "practices": [ "concept/common-root", "concept/discriminant", "concept/multiple-root", "concept/resultant", "method/elimination" ] }, { "id": "dickson-theory-of-equations-1922/ex-page153", "set": "Page153", "page": "153", "chapter": "dickson-theory-of-equations-1922/ch-x", "practices": [ "concept/conic-section", "concept/cubic-equation", "concept/root-of-an-equation", "method/elimination" ] }, { "id": "dickson-theory-of-equations-1922/ex-page49b", "set": "Page49b", "page": "49", "chapter": "dickson-theory-of-equations-1922/ch-iv", "practices": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153b", "set": "Page153b", "page": "153", "chapter": "dickson-theory-of-equations-1922/ch-x", "practices": [ "concept/cubic-equation", "concept/determinant", "concept/discriminant", "concept/resultant" ] } ], "problems": [ { "id": "dickson-theory-of-equations-1922/ex-page10/1", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 1", "problem_latex": "Show that the primitive cube roots of unity are $\\omega$ and~$\\omega^{2}$.", "markdown": "Show that the primitive cube roots of unity are $\\omega$ and $\\omega^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/2", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 2", "problem_latex": "For $R$ given by~\\Eq{7}, prove that the primitive $n$th\nroots of unity are (i)~for $n=6$,\n$R$, $R^5$; (ii)~for $n=8$, $R$, $R^3$, $R^5$, $R^7$; (iii)~for $n=12$, $R$, $R^5$, $R^7$, $R^{11}$.", "markdown": "For $R$ given by $(7)$, prove that the primitive $n$th roots of unity are (i) for $n=6$, $R$, $R^5$; (ii) for $n=8$, $R$, $R^3$, $R^5$, $R^7$; (iii) for $n=12$, $R$, $R^5$, $R^7$, $R^{11}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/3", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 3", "problem_latex": "When $n$ is a prime, prove that any $n$th root of unity, other than~$1$, is primitive.", "markdown": "When $n$ is a prime, prove that any $n$th root of unity, other than $1$, is primitive.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/4", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 4", "problem_latex": "Let $R$ be a primitive $n$th root \\Eq{7} of unity, where $n$ is a product of two different\nprimes $p$ and~$q$. Show that $R, \\dotsc, R^n$ are primitive with the exception of $R^p$, $R^{2p}, \\dotsc,\nR^{qp}$, whose $q$th powers are unity, and $R^q$, $R^{2q}, \\dotsc, R^{pq}$, whose $p$th powers are unity.\nThese two sets of exceptions have only $R^{pq}$ in common. Hence there are exactly\n$pq - p - q + 1$ primitive $n$th roots of unity.", "markdown": "Let $R$ be a primitive $n$th root $(7)$ of unity, where $n$ is a product of two different primes $p$ and $q$. Show that $R, \\dotsc, R^n$ are primitive with the exception of $R^p$, $R^{2p}, \\dotsc, R^{qp}$, whose $q$th powers are unity, and $R^q$, $R^{2q}, \\dotsc, R^{pq}$, whose $p$th powers are unity. These two sets of exceptions have only $R^{pq}$ in common. Hence there are exactly $pq - p - q + 1$ primitive $n$th roots of unity.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/5", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 5", "problem_latex": "Find the number of primitive $n$th roots of unity if $n$ is a square of a prime~$p$.", "markdown": "Find the number of primitive $n$th roots of unity if $n$ is a square of a prime $p$.", "answer_latex": [ "$p(p-1)$." ], "answer_markdown": [ "$p(p-1)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "p*(p-1)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/6", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 6", "problem_latex": "Extend Ex.~4 to the case in which $n$ is a product of three distinct primes.", "markdown": "Extend Ex. 4 to the case in which $n$ is a product of three distinct primes.", "answer_latex": [ "$(p-1)(q-1)(r-1)$ if $n=pqr$." ], "answer_markdown": [ "$(p-1)(q-1)(r-1)$ if $n=pqr$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "(p-1)*(q-1)*(r-1)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/7", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 7", "problem_latex": "If $R$ is a primitive $15$th root \\Eq{7} of unity, verify that $R^3$, $R^6$, $R^9$, $R^{12}$ are the primitive\nfifth roots of unity, and $R^5$ and~$R^{10}$ are the primitive cube roots of unity. Show\nthat their eight products by pairs give all the primitive $15$th roots of unity.", "markdown": "If $R$ is a primitive $15$th root $(7)$ of unity, verify that $R^3$, $R^6$, $R^9$, $R^{12}$ are the primitive fifth roots of unity, and $R^5$ and $R^{10}$ are the primitive cube roots of unity. Show that their eight products by pairs give all the primitive $15$th roots of unity.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/8", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 8", "problem_latex": "If $\\rho$ is any primitive $n$th root of unity, prove that $\\rho$, $\\rho^2, \\dots, \\rho^n$ are distinct and\ngive all the $n$th roots of unity. Of these show that $\\rho^k$ is a primitive $n$th root of unity\nif and only if $k$ is relatively prime to~$n$.", "markdown": "If $\\rho$ is any primitive $n$th root of unity, prove that $\\rho$, $\\rho^2, \\dots, \\rho^n$ are distinct and give all the $n$th roots of unity. Of these show that $\\rho^k$ is a primitive $n$th root of unity if and only if $k$ is relatively prime to $n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page10/9", "set": "dickson-theory-of-equations-1922/ex-page10", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "10", "location": "Exercise Page10, problem 9", "problem_latex": "Show that the six primitive $18$th roots of unity are the negatives of the primitive\nninth roots of unity.", "markdown": "Show that the six primitive $18$th roots of unity are the negatives of the primitive ninth roots of unity.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/1", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 1", "problem_latex": "What arc of a circle is double its chord?", "markdown": "What arc of a circle is double its chord?", "answer_latex": [ "$217° 12' 27.4'' = 3.790988$ radians." ], "answer_markdown": [ "$217° 12' 27.4'' = 3.790988$ radians." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(theta, 4*sin(theta/2))", "answer_expr": "3.790988" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x, 4*sin(x/2))" ], "shape": [ "solve: Eq(x, N*sin(N*x))" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig", "core.units" ], "expectation": "X=3.790988", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/10i", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 10, "part": "i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 10i", "problem_latex": "$4 \\tau x^3 - (3x - 1)^2 = 0$ arises in the study of the isothermals of a gas. Find its\nroots when (i)~$\\tau = 0.002$ and (ii)~$\\tau = 0.99$.", "markdown": "$4 \\tau x^3 - (3x - 1)^2 = 0$ arises in the study of the isothermals of a gas. Find its roots when (i) $\\tau = 0.002$ and (ii) $\\tau = 0.99$.", "answer_latex": [ "(i) $0.327739$, $0.339224$, $1124.333037$. \\\\" ], "answer_markdown": [ "(i) $0.327739$, $0.339224$, $1124.333037$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4*0.002*x**3 - (3*x - 1)**2, 0)", "answer_expr": [ 0.327739, 0.339224, 1124.333037 ] } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3/125 - (3*x - 1)**2, 0)" ], "shape": [ "solve: Eq(N*x**N - (N*x - 1)**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/10ii", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 10, "part": "ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 10ii", "problem_latex": "$4 \\tau x^3 - (3x - 1)^2 = 0$ arises in the study of the isothermals of a gas. Find its\nroots when (i)~$\\tau = 0.002$ and (ii)~$\\tau = 0.99$.", "markdown": "$4 \\tau x^3 - (3x - 1)^2 = 0$ arises in the study of the isothermals of a gas. Find its roots when (i) $\\tau = 0.002$ and (ii) $\\tau = 0.99$.", "answer_latex": [ "(ii) $0.250279$, $0.894609$, $1.127839$. \\hfill\\break\n % [** PP: If \\\\ above, LaTeX thinks next token is an optional argument]\n [Set $x = 1 + y$, $y = 1/z$ and solve by trigonometry.]" ], "answer_markdown": [ "(ii) $0.250279$, $0.894609$, $1.127839$. % [** PP: If above, LaTeX thinks next token is an optional argument] [Set $x = 1 + y$, $y = 1/z$ and solve by trigonometry.]" ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4*0.99*x**3 - (3*x - 1)**2, 0)", "answer_expr": [ 0.250279, 0.894609, 1.127839 ] } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(99*x**3/25 - (3*x - 1)**2, 0)" ], "shape": [ "solve: Eq(N*x**N - (N*x - 1)**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/11", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 11", "problem_latex": "Solve $x^x = 100$.", "markdown": "Solve $x^x = 100$.", "answer_latex": [ "$3.597285$." ], "answer_markdown": [ "$3.597285$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**x, 100)", "answer_expr": "3.597285" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**x, 100)" ], "shape": [ "solve: Eq(x**x, N)" ], "same_problem_in": [], "needs": [ "core.log", "core.solve.num" ], "expectation": "X=3.597285", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/12", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 12", "problem_latex": "Solve $x = 10\\log x$.", "markdown": "Solve $x = 10\\log x$.", "answer_latex": [ "$10$, $1.371288$." ], "answer_markdown": [ "$10$, $1.371288$." ], "checks": [ { "task": "solve", "verdict": "FLAG-PARSE", "judge_why": "ValueError: Exceeds the limit (4300 digits) for integer string conversion; use sys.set_int_max_str_digits() to increase the limit", "problem_expr": "Eq(x, 10*log(x, 10))", "answer_expr": [ 10, 1.371288 ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "solve: Eq(x, 10*log(x)/log(10))" ], "shape": [ "solve: Eq(x, N*log(x)/log(N))" ], "same_problem_in": [], "needs": [ "core.log", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/13", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 13, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 13", "problem_latex": "Solve $x + \\log x = x \\log x$.", "markdown": "Solve $x + \\log x = x \\log x$.", "answer_latex": [ "$0.326878$, $12.267305$." ], "answer_markdown": [ "$0.326878$, $12.267305$." ], "checks": [ { "task": "solve", "verdict": "FLAG-PARSE", "judge_why": "ValueError: Exceeds the limit (4300 digits) for integer string conversion; use sys.set_int_max_str_digits() to increase the limit", "problem_expr": "Eq(x + log(x, 10), x*log(x, 10))", "answer_expr": [ 0.326878, 12.267305 ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "solve: Eq(x + log(x)/log(10), x*log(x)/log(10))" ], "shape": [ "solve: Eq(x + log(x)/log(N), x*log(x)/log(N))" ], "same_problem_in": [], "needs": [ "core.log", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/14", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 14", "problem_latex": "Solve Kepler's equation $M = x - e \\sin x$ when $M = 332° 28' 54.8''$,\n $e = 14° 3' 20''$.", "markdown": "Solve Kepler’s equation $M = x - e \\sin x$ when $M = 332° 28' 54.8''$, $e = 14° 3' 20''$.", "answer_latex": [ "$324° 16' 29.55''$." ], "answer_markdown": [ "$324° 16' 29.55''$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x - 253*pi*sin(x)/3240, 2992337*pi/1620000) fails at {x: 864733*pi/480000}: leaves -2.34553721304E-7", "problem_expr": "Eq(x - (14 + 3/60 + 20/3600)*pi/180*sin(x), (332 + 28/60 + 54.8/3600)*pi/180)", "answer_expr": "(324 + 16/60 + 29.55/3600)*pi/180" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x - 253*pi*sin(x)/3240, 2992337*pi/1620000)" ], "shape": [ "solve: Eq(pi*N*sin(x) + x, pi*N)" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/15", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 15, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 15", "problem_latex": "In what time would a sum of money at 6\\% interest compounded annually\namount to as much as the same sum at simple interest at~8\\%?", "markdown": "In what time would a sum of money at 6% interest compounded annually amount to as much as the same sum at simple interest at 8%?", "answer_latex": [ "$10$~yr.\\ $4$~mo.\\ $0$~days." ], "answer_markdown": [ "$10$ yr. $4$ mo. $0$ days." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "t": 10.333406 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(524662411096539147*x/4882812500000000000 + 174887470365513049/244140625000000000, 2*x/25 + 1)" ], "shape": [ "solve: Eq(N*x + N, N*x + 1)" ], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/16", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 16, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 16", "problem_latex": "In a semicircle of diameter~$x$ is inscribed a quadrilateral with sides $a$, $b$, $c$,~$x$;\nthen $x^3 - (a^2 + b^2 + c^2) x - 2abc = 0$ (I.~Newton). Given $a = 2$, $b = 3$, $c = 4$, find~$x$.", "markdown": "In a semicircle of diameter $x$ is inscribed a quadrilateral with sides $a$, $b$, $c$, $x$; then $x^3 - (a^2 + b^2 + c^2) x - 2abc = 0$ (I. Newton). Given $a = 2$, $b = 3$, $c = 4$, find $x$.", "answer_latex": [ "$6.074674$." ], "answer_markdown": [ "$6.074674$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - (2**2 + 3**2 + 4**2)*x - 2*2*3*4, 0)", "answer_expr": "6.074674" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 29*x - 48, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X=6.074674", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/17", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 17, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 17", "problem_latex": "What rate of interest is implied in an offer to sell a house for \\$9000~cash, or\n\\$1000~down and \\$3000 at the end of each year for three years?", "markdown": "What rate of interest is implied in an offer to sell a house for $9000 cash, or $1000 down and $3000 at the end of each year for three years?", "answer_latex": [ "$6.13$\\%." ], "answer_markdown": [ "$6.13$%." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(3000*(1 - (1 + r)**(-3))/r, 8000)", "answer_expr": "0.0613" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq((3000 - 3000/(x + 1)**3)/x, 8000)" ], "shape": [ "solve: Eq((N*(x + 1)**N + N)/x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num" ], "expectation": "X=0.0613", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/2", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 2", "problem_latex": "What arc of a circle is double the distance from the center of the circle to the\nchord of the arc?", "markdown": "What arc of a circle is double the distance from the center of the circle to the chord of the arc?", "answer_latex": [ "$42° 20' 47\\tfrac{1}{4}''$ doubled." ], "answer_markdown": [ "$42° 20' 47\\tfrac{1}{4}''$ doubled." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(theta, 2*cos(theta/2)) fails at {theta: 203263*pi/432000}: leaves -2.92544153125E-8", "problem_expr": "Eq(theta, 2*cos(theta/2))", "answer_expr": "2*(42 + 20/60 + 47.25/3600)*pi/180" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x, 2*cos(x/2))" ], "shape": [ "solve: Eq(x, N*cos(N*x))" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/3", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 3", "problem_latex": "If $A$ and~$B$ are the points of contact of two tangents to a circle of radius unity\nfrom a point~$P$ without it, and if arc $AB$ is equal to~$PA$, find the length of the arc.", "markdown": "If $A$ and $B$ are the points of contact of two tangents to a circle of radius unity from a point $P$ without it, and if arc $AB$ is equal to $PA$, find the length of the arc.", "answer_latex": [ "$133° 33.8'$." ], "answer_markdown": [ "$133° 33.8'$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(theta, tan(theta/2)) fails at {theta: 40069*pi/54000}: leaves 0.00000541806298619", "problem_expr": "Eq(theta, tan(theta/2))", "answer_expr": "(133 + 33.8/60)*pi/180" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x, tan(x/2))" ], "shape": [ "solve: Eq(x, tan(N*x))" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/4", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 4", "problem_latex": "Find the angle at the center of a circle of a sector which is bisected by its chord.", "markdown": "Find the angle at the center of a circle of a sector which is bisected by its chord.", "answer_latex": [ "$108° 36' 14''$." ], "answer_markdown": [ "$108° 36' 14''$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(sin(theta), theta/2) fails at {theta: 195487*pi/324000}: leaves -9.61981279563E-7", "problem_expr": "Eq(sin(theta), theta/2)", "answer_expr": "(108 + 36/60 + 14/3600)*pi/180" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(sin(x), x/2)" ], "shape": [ "solve: Eq(sin(x), N*x)" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/5", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 5", "problem_latex": "Find the radius of the smallest hollow iron sphere, with air exhausted, which will\nfloat in water if its shell is $1$~inch thick and the specific gravity of iron is~$7.5$.", "markdown": "Find the radius of the smallest hollow iron sphere, with air exhausted, which will float in water if its shell is $1$ inch thick and the specific gravity of iron is $7.5$.", "answer_latex": [ "$21.468212$." ], "answer_markdown": [ "$21.468212$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "R": 21.468212 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x**3, 15*x**3/2 - 15*(x - 1)**3/2)" ], "shape": [ "solve: Eq(x**N, N*x**N + N*(x - 1)**N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/6", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 6", "problem_latex": "From one end of a diameter of a circle draw a chord which bisects the semicircle.", "markdown": "From one end of a diameter of a circle draw a chord which bisects the semicircle.", "answer_latex": [ "Angle at center $47° 39' 13''$." ], "answer_markdown": [ "Angle at center $47° 39' 13''$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/7", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 7", "problem_latex": "The equation $x \\tan x = c$ occurs in the theory of vibrating strings. Its approximate\nsolutions may be found from the graphs $y = \\cot x$, $y = x/c$. Find $x$ when $c = 1$.", "markdown": "The equation $x \\tan x = c$ occurs in the theory of vibrating strings. Its approximate solutions may be found from the graphs $y = \\cot x$, $y = x/c$. Find $x$ when $c = 1$.", "answer_latex": [ "$49° 17' 36.5''$." ], "answer_markdown": [ "$49° 17' 36.5''$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x*tan(x), 1) fails at {x: 354913*pi/1296000}: leaves -6.33821388201E-7", "problem_expr": "Eq(x*tan(x), 1)", "answer_expr": "(49 + 17/60 + 36.5/3600)*pi/180" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x*tan(x), 1)" ], "shape": [ "solve: Eq(x*tan(x), 1)" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/8", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 8", "problem_latex": "The equation $\\tan x = x$ occurs in the study of the vibrations of air in a spherical\ncavity. From an approximate solution $x_1 = 1.5\\pi$, we obtain successively better approximations\n$x_2 = \\tan^{-1} x_1 = 1.4334 \\pi$, $x_3 = \\tan^{-1} x_2, \\dotsc$. Find the first three solutions to\n4~decimal places.", "markdown": "The equation $\\tan x = x$ occurs in the study of the vibrations of air in a spherical cavity. From an approximate solution $x_1 = 1.5\\pi$, we obtain successively better approximations $x_2 = \\tan^{-1} x_1 = 1.4334 \\pi$, $x_3 = \\tan^{-1} x_2, \\dotsc$. Find the first three solutions to 4 decimal places.", "answer_latex": [ "$1.4303\\pi$, $2.4590\\pi$, $3.4709\\pi$;\\quad $257° 27' 12.225''$ more exact than first." ], "answer_markdown": [ "$1.4303\\pi$, $2.4590\\pi$, $3.4709\\pi$; $257° 27' 12.225''$ more exact than first." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(tan(x), x) fails at {x: 14303*pi/10000}: leaves 0.000212306355056", "problem_expr": "Eq(tan(x), x)", "answer_expr": [ "1.4303*pi", "2.4590*pi", "3.4709*pi" ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(tan(x), x)" ], "shape": [ "solve: Eq(tan(x), x)" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page100/9", "set": "dickson-theory-of-equations-1922/ex-page100", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "100", "location": "Exercise Page100, problem 9", "problem_latex": "Find to 3~decimal places the first five solutions of\n\\[\n\\tan x = \\frac{2x}{2-x^2},\n\\]\nwhich occurs in the theory of vibrations in a conical pipe.", "markdown": "Find to 3 decimal places the first five solutions of x = 2x2-x^2, which occurs in the theory of vibrations in a conical pipe.", "answer_latex": [ "$x/\\pi = 0.6625, 1.891, 2.930, 3.948, 4.959$." ], "answer_markdown": [ "$x/\\pi = 0.6625, 1.891, 2.930, 3.948, 4.959$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(tan(x), 2*x/(2 - x**2)) fails at {x: 53*pi/80}: leaves -0.000503334238641", "problem_expr": "Eq(tan(x), 2*x/(2 - x**2))", "answer_expr": [ "0.6625*pi", "1.891*pi", "2.930*pi", "3.948*pi", "4.959*pi" ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(tan(x), 2*x/(-x**2 + 2))" ], "shape": [ "solve: Eq(tan(x), N*x/(N - x**N))" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page102/1", "set": "dickson-theory-of-equations-1922/ex-page102", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "102", "location": "Exercise Page102, problem 1", "problem_latex": "$\\begin{System}[\\,]{2}\n 8x &-{}& y &= 34, \\\\\n x &+{}& 8y &= 53.\n\\end{System}$", "markdown": "$\\begin{System}[\\,]{2} 8x &-{}& y &= 34, \\\\ x &+{}& 8y &= 53. \\end{System}$", "answer_latex": [ "$x = 5$, $y = 6$." ], "answer_markdown": [ "$x = 5$, $y = 6$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 5, y: 6}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(8*a + x, 53), Eq(-a + 8*x, 34))" ], "shape": [ "solve: (Eq(N*a + x, N), Eq(N*x - a, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page102/2", "set": "dickson-theory-of-equations-1922/ex-page102", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "102", "location": "Exercise Page102, problem 2", "problem_latex": "$\\begin{System}[\\,]{2}\n 3x &+{}& 4y &= 10, \\\\\n 4x &+{}& y &= 9.\n\\end{System}$", "markdown": "$\\begin{System}[\\,]{2} 3x &+{}& 4y &= 10, \\\\ 4x &+{}& y &= 9. \\end{System}$", "answer_latex": [ "$x = 2$, $y = 1$." ], "answer_markdown": [ "$x = 2$, $y = 1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 2, y: 1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(4*a + 3*x, 10), Eq(a + 4*x, 9))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*x + a, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page102/3", "set": "dickson-theory-of-equations-1922/ex-page102", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "102", "location": "Exercise Page102, problem 3", "problem_latex": "$\\begin{System}[\\,]{2}\n ax &+{}& by &= a^2, \\\\\n bx &-{}& ay &= ab.\n\\end{System}$", "markdown": "$\\begin{System}[\\,]{2} ax &+{}& by &= a^2, \\\\ bx &-{}& ay &= ab. \\end{System}$", "answer_latex": [ "$x = a$, $y = 0$." ], "answer_markdown": [ "$x = a$, $y = 0$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: a, y: 0}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a*x + b*c, a**2), Eq(-a*c + b*x, a*b))" ], "shape": [ "solve: (Eq(a*x + b*c, a**N), Eq(-a*c + b*x, a*b))" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page106/1", "set": "dickson-theory-of-equations-1922/ex-page106", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "106", "location": "Exercise Page106, problem 1", "problem_latex": "Find the six terms involving $a_2$ in the determinant~\\Eq{7}.", "markdown": "Find the six terms involving $a_2$ in the determinant $(7)$.", "answer_latex": [ "$-a_2b_1c_3d_4 + a_2b_1c_4d_3 + a_2b_3c_1d_4\n - a_2b_3c_4d_1 - a_2b_4c_1d_3 + a_2b_4c_3d_1$." ], "answer_markdown": [ "$-a_2b_1c_3d_4 + a_2b_1c_4d_3 + a_2b_3c_1d_4 - a_2b_3c_4d_1 - a_2b_4c_1d_3 + a_2b_4c_3d_1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page106/2", "set": "dickson-theory-of-equations-1922/ex-page106", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "106", "location": "Exercise Page106, problem 2", "problem_latex": "What are the signs of $a_3b_5c_2d_1e_4$, $a_5b_4c_3d_2e_1$ in a determinant of order five?", "markdown": "What are the signs of $a_3b_5c_2d_1e_4$, $a_5b_4c_3d_2e_1$ in a determinant of order five?", "answer_latex": [ "$+$, $+$." ], "answer_markdown": [ "$+$, $+$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page106/3", "set": "dickson-theory-of-equations-1922/ex-page106", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "106", "location": "Exercise Page106, problem 3", "problem_latex": "Show that the arrangement $4, 1, 3, 2$ may be obtained from $1, 2, 3, 4$ by use of\nthe two successive interchanges $(1, 4)$, $(1, 2)$, and also by use of the four successive\ninterchanges $(1, 4)$, $(1, 3)$, $(1, 2)$, $(2, 3)$.", "markdown": "Show that the arrangement $4, 1, 3, 2$ may be obtained from $1, 2, 3, 4$ by use of the two successive interchanges $(1, 4)$, $(1, 2)$, and also by use of the four successive interchanges $(1, 4)$, $(1, 3)$, $(1, 2)$, $(2, 3)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page106/4", "set": "dickson-theory-of-equations-1922/ex-page106", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "106", "location": "Exercise Page106, problem 4", "problem_latex": "Write out the six terms of~\\Eq{8} for $n = 3$, rearrange the factors of each term so that\nthe new first subscripts shall be in the order $1, 2, 3$, and verify that the resulting six\nterms are those of the determinant~$D'$ in~§85 for $n = 3$.", "markdown": "Write out the six terms of $(8)$ for $n = 3$, rearrange the factors of each term so that the new first subscripts shall be in the order $1, 2, 3$, and verify that the resulting six terms are those of the determinant $D'$ in §85 for $n = 3$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page112/1", "set": "dickson-theory-of-equations-1922/ex-page112", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "112", "location": "Exercise Page112, problem 1", "problem_latex": "$\\ds\n\\begin{vmatrix}\n3a & 3b & 3c \\\\\n5a & 5b & 5c \\\\\n d & e & f\n\\end{vmatrix} = 0$.", "markdown": "$\\ds \\begin{vmatrix} 3a & 3b & 3c \\\\ 5a & 5b & 5c \\\\ d & e & f \\end{vmatrix} = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page112/2", "set": "dickson-theory-of-equations-1922/ex-page112", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "112", "location": "Exercise Page112, problem 2", "problem_latex": "$\\ds\n\\begin{vmatrix}\n2r & l & 3r \\\\\n2s & m & 3s \\\\\n2t & n & 3t\n\\end{vmatrix} = 0$.", "markdown": "$\\ds \\begin{vmatrix} 2r & l & 3r \\\\ 2s & m & 3s \\\\ 2t & n & 3t \\end{vmatrix} = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page112/3", "set": "dickson-theory-of-equations-1922/ex-page112", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "112", "location": "Exercise Page112, problem 3", "problem_latex": "$\\ds\n\\begin{vmatrix}\n2 & 7 & 3 \\\\\n5 & 9 & 8 \\\\\n0 & 3 & 0\n\\end{vmatrix}$.", "markdown": "$\\ds \\begin{vmatrix} 2 & 7 & 3 \\\\ 5 & 9 & 8 \\\\ 0 & 3 & 0 \\end{vmatrix}$.", "answer_latex": [ "$-3$." ], "answer_markdown": [ "$-3$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -3.0, printed -3 (half-unit 0.5; correctly rounded at the printed digits: -3.0)", "problem_expr": "Matrix([[2, 7, 3], [5, 9, 8], [0, 3, 0]]).det()", "answer_expr": "-3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -3" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.matrix" ], "expectation": "X#-3,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 8 × 3 ENTER 5 × − 3 × +/−" ], "constants": [ { "value": "2", "source": "the (1,1) entry of the matrix, the 2 in row 1 column 1" }, { "value": "8", "source": "the (2,3) entry of the matrix, the 8 in row 2 column 3" }, { "value": "3", "source": "the (1,3) entry of the matrix, the 3 in row 1 column 3" }, { "value": "5", "source": "the (2,1) entry of the matrix, the 5 in row 2 column 1" }, { "value": "3", "source": "the (3,2) entry of the matrix, the 3 in row 3 column 2, the only nonzero entry of row 3 besides the zeros" } ], "calculator_value": "-3E+0", "printed_value": "-3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page112/4", "set": "dickson-theory-of-equations-1922/ex-page112", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "112", "location": "Exercise Page112, problem 4", "problem_latex": "$\\ds\n\\begin{vmatrix}\n5 & 7 & 0 \\\\\n6 & 8 & 0 \\\\\n3 & 9 & 4\n\\end{vmatrix}$.", "markdown": "$\\ds \\begin{vmatrix} 5 & 7 & 0 \\\\ 6 & 8 & 0 \\\\ 3 & 9 & 4 \\end{vmatrix}$.", "answer_latex": [ "$-8$." ], "answer_markdown": [ "$-8$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -8.0, printed -8 (half-unit 0.5; correctly rounded at the printed digits: -8.0)", "problem_expr": "Matrix([[5, 7, 0], [6, 8, 0], [3, 9, 4]]).det()", "answer_expr": "-8" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -8" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.matrix" ], "expectation": "X#-8,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 8 ×", "7 ENTER 6 ×", "−", "4 ×" ], "calculator_value": "-8E+0", "printed_value": "-8", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page112/5", "set": "dickson-theory-of-equations-1922/ex-page112", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "112", "location": "Exercise Page112, problem 5", "problem_latex": "$\\ds\n\\begin{vmatrix}\na & b & c & d \\\\\na^2 & b^2 & c^2 & d^2 \\\\\na^3 & b^3 & c^3 & d^3 \\\\\na^4 & b^4 & c^4 & d^4\n\\end{vmatrix}\n= abcd(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$.", "markdown": "$\\ds \\begin{vmatrix} a & b & c & d \\\\ a^2 & b^2 & c^2 & d^2 \\\\ a^3 & b^3 & c^3 & d^3 \\\\ a^4 & b^4 & c^4 & d^4 \\end{vmatrix} = abcd(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115/1", "set": "dickson-theory-of-equations-1922/ex-page115", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Exercise Page115, problem 1", "problem_latex": "$\\begin{System}{3}\n x &+{}& y &+{}& z &= 11, \\\\\n 2x &-{}& 6y &-{}& z &= 0, \\\\\n 3x &+{}& 4y &+{}& 2z &= 0.\n\\end{System}$", "markdown": "$\\begin{System}{3} x &+{}& y &+{}& z &= 11, \\\\ 2x &-{}& 6y &-{}& z &= 0, \\\\ 3x &+{}& 4y &+{}& 2z &= 0. \\end{System}$", "answer_latex": [ "$x = -8$, $y = -7$, $z = 26$." ], "answer_markdown": [ "$x = -8$, $y = -7$, $z = 26$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: -8, y: -7, z: 26}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a + b + c, 11), Eq(2*a - 6*b - c, 0), Eq(3*a + 4*b + 2*c, 0))" ], "shape": [ "solve: (Eq(a + b + c, N), Eq(N*a + N*b - c, 0), Eq(N*a + N*b + N*c, 0))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115/2", "set": "dickson-theory-of-equations-1922/ex-page115", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Exercise Page115, problem 2", "problem_latex": "$\\begin{System}{3}\n x &+{}& y &+{}& z &= 0, \\\\\n x &+{}& 2y &+{}& 3z &= -1, \\\\\n x &+{}& 3y &+{}& 6z &= 0.\n\\end{System}$", "markdown": "$\\begin{System}{3} x &+{}& y &+{}& z &= 0, \\\\ x &+{}& 2y &+{}& 3z &= -1, \\\\ x &+{}& 3y &+{}& 6z &= 0. \\end{System}$", "answer_latex": [ "$x = 3$, $y = -5$, $z = 2$." ], "answer_markdown": [ "$x = 3$, $y = -5$, $z = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: -5, z: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a + b + c, 0), Eq(a + 2*b + 3*c, -1), Eq(a + 3*b + 6*c, 0))" ], "shape": [ "solve: (Eq(a + b + c, 0), Eq(N*b + N*c + a, -1), Eq(N*b + N*c + a, 0))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115/3", "set": "dickson-theory-of-equations-1922/ex-page115", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Exercise Page115, problem 3", "problem_latex": "$\\begin{System}{3}\n x &-{}& 2y &+{}& z &= 12, \\\\\n x &+{}& 2y &+{}& 3z &= 48, \\\\\n6x &+{}& 4y &+{}& 3z &= 84.\n\\end{System}$", "markdown": "$\\begin{System}{3} x &-{}& 2y &+{}& z &= 12, \\\\ x &+{}& 2y &+{}& 3z &= 48, \\\\ 6x &+{}& 4y &+{}& 3z &= 84. \\end{System}$", "answer_latex": [ "$x = 6$, $y = 3$, $z = 12$." ], "answer_markdown": [ "$x = 6$, $y = 3$, $z = 12$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 6, y: 3, z: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a - 2*b + c, 12), Eq(a + 2*b + 3*c, 48), Eq(6*a + 4*b + 3*c, 84))" ], "shape": [ "solve: (Eq(N*b + a + c, N), Eq(N*b + N*c + a, N), Eq(N*a + N*b + N*c, N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115/4", "set": "dickson-theory-of-equations-1922/ex-page115", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Exercise Page115, problem 4", "problem_latex": "$\\begin{System}{2}\n3x &-{}& 2y &= 7, \\\\\n3y &-{}& 2z &= 6, \\\\\n3z &-{}& 2x &= -1.\n\\end{System}$", "markdown": "$\\begin{System}{2} 3x &-{}& 2y &= 7, \\\\ 3y &-{}& 2z &= 6, \\\\ 3z &-{}& 2x &= -1. \\end{System}$", "answer_latex": [ "$x = 5$, $y = 4$, $z = 3$." ], "answer_markdown": [ "$x = 5$, $y = 4$, $z = 3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 5, y: 4, z: 3}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-2*a + 3*c, -1), Eq(3*a - 2*b, 7), Eq(3*b - 2*c, 6))" ], "shape": [ "solve: (Eq(N*a + N*c, -1), Eq(N*a + N*b, N), Eq(N*b + N*c, N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115/5", "set": "dickson-theory-of-equations-1922/ex-page115", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Exercise Page115, problem 5", "problem_latex": "$\\begin{System}{4}\nx &+{}& y &+{}& z &+{}& w &= 1, \\\\\nx &+{}& 2y &+{}& 3z &+{}& 4w &= 11, \\\\\nx &+{}& 3y &+{}& 6z &+{}& 10w &= 26, \\\\\nx &+{}& 4y &+{}& 10z &+{}& 20w &= 47.\n\\end{System}$", "markdown": "$\\begin{System}{4} x &+{}& y &+{}& z &+{}& w &= 1, \\\\ x &+{}& 2y &+{}& 3z &+{}& 4w &= 11, \\\\ x &+{}& 3y &+{}& 6z &+{}& 10w &= 26, \\\\ x &+{}& 4y &+{}& 10z &+{}& 20w &= 47. \\end{System}$", "answer_latex": [ "$x = -5$, $y = 3$, $z = 2$, $w = 1$." ], "answer_markdown": [ "$x = -5$, $y = 3$, $z = 2$, $w = 1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: -5, y: 3, z: 2, w: 1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a + b + c + d, 1), Eq(4*a + b + 2*c + 3*d, 11), Eq(10*a + b + 3*c + 6*d, 26), Eq(20*a + b + 4*c + 10*d, 47))" ], "shape": [ "solve: (Eq(a + b + c + d, 1), Eq(N*a + N*c + N*d + b, N), Eq(N*a + N*c + N*d + b, N), Eq(N*a + N*c + N*d + b, N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115/6", "set": "dickson-theory-of-equations-1922/ex-page115", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Exercise Page115, problem 6", "problem_latex": "$\\begin{System}{4}\n2x &-{}& y &+{}& 3z &-{}& 2w &= 4, \\\\\n x &+{}& 7y &+{}& z &-{}& w &= 2, \\\\\n3x &+{}& 5y &-{}& 5z &+{}& 3w &= 0, \\\\\n4x &-{}& 3y &+{}& 2z &-{}& w &= 5.\n\\end{System}$", "markdown": "$\\begin{System}{4} 2x &-{}& y &+{}& 3z &-{}& 2w &= 4, \\\\ x &+{}& 7y &+{}& z &-{}& w &= 2, \\\\ 3x &+{}& 5y &-{}& 5z &+{}& 3w &= 0, \\\\ 4x &-{}& 3y &+{}& 2z &-{}& w &= 5. \\end{System}$", "answer_latex": [ "$x = 1$, $y = z = 0$, $w = -1$." ], "answer_markdown": [ "$x = 1$, $y = z = 0$, $w = -1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 1, y: 0, z: 0, w: -1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-2*a + 2*b - c + 3*d, 4), Eq(-a + b + 7*c + d, 2), Eq(-a + 4*b - 3*c + 2*d, 5), Eq(3*a + 3*b + 5*c - 5*d, 0))" ], "shape": [ "solve: (Eq(N*a + N*b + N*d - c, N), Eq(N*c - a + b + d, N), Eq(N*b + N*c + N*d - a, N), Eq(N*a + N*b + N*c + N*d, 0))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page115/7", "set": "dickson-theory-of-equations-1922/ex-page115", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "115", "location": "Exercise Page115, problem 7", "problem_latex": "Prove the first relation~\\Eq{13} by multiplying the members of the first equation~\\Eq{12}\nby~$A_{11}$, those of the second equation by $-A_{21}, \\dotsc$, those of the $n$th equation by\n$(-1)^{n-1}A_{n1}$, and adding, where $A_{ij}$ by denotes the minor of~$a_{ij}$ in~$D$. Hint: The resulting\ncoefficient of~$x_2$ is the expansion, according to the elements of its first column, of a determinant\nderived from $D$ by replacing $a_{11}$ by~$a_{12}$, $\\dotsc$, $a_{n1}$ by~$a_{n2}$.", "markdown": "Prove the first relation $(13)$ by multiplying the members of the first equation $(12)$ by $A_{11}$, those of the second equation by $-A_{21}, \\dotsc$, those of the $n$th equation by $(-1)^{n-1}A_{n1}$, and adding, where $A_{ij}$ by denotes the minor of $a_{ij}$ in $D$. Hint: The resulting coefficient of $x_2$ is the expansion, according to the elements of its first column, of a determinant derived from $D$ by replacing $a_{11}$ by $a_{12}$, $\\dotsc$, $a_{n1}$ by $a_{n2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/1", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 1", "problem_latex": "$\\begin{System}{3}\n2x&+{}& y&+{}& 3z &= 1, \\\\\n4x&+{}& 2y&-{}& z &= -3, \\\\\n2x&+{}& y&-{}& 4z &= -4.\n\\end{System}$", "markdown": "$\\begin{System}{3} 2x&+{}& y&+{}& 3z &= 1, \\\\ 4x&+{}& 2y&-{}& z &= -3, \\\\ 2x&+{}& y&-{}& 4z &= -4. \\end{System}$", "answer_latex": [ "Consistent: $y = -8/7 - 2x$, $z = 5/7$ (common line)." ], "answer_markdown": [ "Consistent: $y = -8/7 - 2x$, $z = 5/7$ (common line)." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/2", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 2", "problem_latex": "$\\begin{System}{3}\n2x&+{}& y&+{}& 3z &= 1, \\\\\n4x&+{}& 2y&-{}& z &= 3, \\\\\n2x&+{}& y&-{}& 4z &= 4.\n\\end{System}$", "markdown": "$\\begin{System}{3} 2x&+{}& y&+{}& 3z &= 1, \\\\ 4x&+{}& 2y&-{}& z &= 3, \\\\ 2x&+{}& y&-{}& 4z &= 4. \\end{System}$", "answer_latex": [ "Inconsistent, case $(\\beta)$." ], "answer_markdown": [ "Inconsistent, case $(\\beta)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/3", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 3", "problem_latex": "$\\begin{System}{3}\n x&-{}& 3y&+{}& 4z &= 1, \\\\\n4x&-{}& 12y&+{}& 16z &= 3, \\\\\n3x&-{}& 9y&+{}& 12z &= 3.\n\\end{System}$", "markdown": "$\\begin{System}{3} x&-{}& 3y&+{}& 4z &= 1, \\\\ 4x&-{}& 12y&+{}& 16z &= 3, \\\\ 3x&-{}& 9y&+{}& 12z &= 3. \\end{System}$", "answer_latex": [ "Inconsistent (two parallel planes)." ], "answer_markdown": [ "Inconsistent (two parallel planes)." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/4", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 4", "problem_latex": "$\\begin{System}{3}\n x&-{}& 3y&+{}& 4z &= 1, \\\\\n4x&-{}& 12y&+{}& 16z &= 4, \\\\\n3x&-{}& 9y&+{}& 12z &= 3.\n\\end{System}$", "markdown": "$\\begin{System}{3} x&-{}& 3y&+{}& 4z &= 1, \\\\ 4x&-{}& 12y&+{}& 16z &= 4, \\\\ 3x&-{}& 9y&+{}& 12z &= 3. \\end{System}$", "answer_latex": [ "Consistent (single plane)." ], "answer_markdown": [ "Consistent (single plane)." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/5a", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 5, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 5a", "problem_latex": "Discuss the system\n\\[\n\\begin{System}{3}\nax&+{}& y&+{}& z &= a-3, \\\\\n x&+{}& ay&+{}& z &= -2, \\\\\n x&+{}& y&+{}& az &= -2,\n\\end{System}\n\\]\nwhen (\\emph{i})~$a = 1$; (\\emph{ii})~$a = -2$; (\\emph{iii})~$a \\neq 1$, $-2$, obtaining the simplest forms of the\nunknowns.", "markdown": "Discuss the system System3 ax&+& y&+& z &= a-3, x&+& ay&+& z &= -2, x&+& y&+& az &= -2, System when (*i*) $a = 1$; (*ii*) $a = -2$; (*iii*) $a \\neq 1$, $-2$, obtaining the simplest forms of the unknowns.", "answer_latex": [ "$z = -x-y-2$. \\hfill" ], "answer_markdown": [ "$z = -x-y-2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/5b", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 5, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 5b", "problem_latex": "Discuss the system\n\\[\n\\begin{System}{3}\nax&+{}& y&+{}& z &= a-3, \\\\\n x&+{}& ay&+{}& z &= -2, \\\\\n x&+{}& y&+{}& az &= -2,\n\\end{System}\n\\]\nwhen (\\emph{i})~$a = 1$; (\\emph{ii})~$a = -2$; (\\emph{iii})~$a \\neq 1$, $-2$, obtaining the simplest forms of the\nunknowns.", "markdown": "Discuss the system System3 ax&+& y&+& z &= a-3, x&+& ay&+& z &= -2, x&+& y&+& az &= -2, System when (*i*) $a = 1$; (*ii*) $a = -2$; (*iii*) $a \\neq 1$, $-2$, obtaining the simplest forms of the unknowns.", "answer_latex": [ "inconsistent. \\hfill" ], "answer_markdown": [ "inconsistent." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/5c", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 5, "part": "c", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 5c", "problem_latex": "Discuss the system\n\\[\n\\begin{System}{3}\nax&+{}& y&+{}& z &= a-3, \\\\\n x&+{}& ay&+{}& z &= -2, \\\\\n x&+{}& y&+{}& az &= -2,\n\\end{System}\n\\]\nwhen (\\emph{i})~$a = 1$; (\\emph{ii})~$a = -2$; (\\emph{iii})~$a \\neq 1$, $-2$, obtaining the simplest forms of the\nunknowns.", "markdown": "Discuss the system System3 ax&+& y&+& z &= a-3, x&+& ay&+& z &= -2, x&+& y&+& az &= -2, System when (*i*) $a = 1$; (*ii*) $a = -2$; (*iii*) $a \\neq 1$, $-2$, obtaining the simplest forms of the unknowns.", "answer_latex": [ "$x = \\dfrac{a - 1}{a + 2}$, $y = z =\\dfrac{-3}{a + 2}$." ], "answer_markdown": [ "$x = \\dfrac{a - 1}{a + 2}$, $y = z =\\dfrac{-3}{a + 2}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: (a - 1)/(a + 2), y: -3/(a + 2), z: -3/(a + 2)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a*x + b + c, a - 3), Eq(a*b + c + x, -2), Eq(a*c + b + x, -2))" ], "shape": [ "solve: (Eq(a*x + b + c, N + a), Eq(a*b + c + x, N), Eq(a*c + b + x, N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/6a", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 6, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 6a", "problem_latex": "Discuss the system\n\\[\n\\begin{System}{3}\n x &+{}& y&+{}& z &= 1, \\\\\nax &+{}& by&+{}& cz &= k, \\\\\na^2x&+{}& b^2y&+{}& c^2z &= k^2,\n\\end{System}\n\\]\nwhen (\\emph{i})~$a$, $b$, $c$ are distinct; (\\emph{ii}) $a = b \\neq c$; (\\emph{iii}) $a = b = c$.", "markdown": "Discuss the system System3 x &+& y&+& z &= 1, ax &+& by&+& cz &= k, a^2x&+& b^2y&+& c^2z &= k^2, System when (*i*) $a$, $b$, $c$ are distinct; (*ii*) $a = b \\neq c$; (*iii*) $a = b = c$.", "answer_latex": [ "$x = \\dfrac{(k-b)(c-k)}{(a-b)(c-a)}$." ], "answer_markdown": [ "$x = \\dfrac{(k-b)(c-k)}{(a-b)(c-a)}$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x + y + z, 1) fails at {x: (-b + k)*(c - k)/((-a + c)*(a - b))}: ('2.8033887323850577267', '0.0')", "problem_expr": null, "answer_expr": "{x: (k-b)*(c-k)/((a-b)*(c-a))}" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: (Eq(e + f + x, 1), Eq(a*x + b*e + c*f, d), Eq(a**2*x + b**2*e + c**2*f, d**2))" ], "shape": [ "solve: (Eq(e + f + x, 1), Eq(a*x + b*e + c*f, d), Eq(a**N*x + b**N*e + c**N*f, d**N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/6b", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 6, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 6b", "problem_latex": "Discuss the system\n\\[\n\\begin{System}{3}\n x &+{}& y&+{}& z &= 1, \\\\\nax &+{}& by&+{}& cz &= k, \\\\\na^2x&+{}& b^2y&+{}& c^2z &= k^2,\n\\end{System}\n\\]\nwhen (\\emph{i})~$a$, $b$, $c$ are distinct; (\\emph{ii}) $a = b \\neq c$; (\\emph{iii}) $a = b = c$.", "markdown": "Discuss the system System3 x &+& y&+& z &= 1, ax &+& by&+& cz &= k, a^2x&+& b^2y&+& c^2z &= k^2, System when (*i*) $a$, $b$, $c$ are distinct; (*ii*) $a = b \\neq c$; (*iii*) $a = b = c$.", "answer_latex": [ "$y = \\dfrac{k-c}{a-c}-x$,\n $z = \\dfrac{a-k}{a-c}$ if $k=a$ or $k=c$, but\n inconsistent if $k$ is different from $a$ and~$c$." ], "answer_markdown": [ "$y = \\dfrac{k-c}{a-c}-x$, $z = \\dfrac{a-k}{a-c}$ if $k=a$ or $k=c$, but inconsistent if $k$ is different from $a$ and $c$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page119/6c", "set": "dickson-theory-of-equations-1922/ex-page119", "number": 6, "part": "c", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "119", "location": "Exercise Page119, problem 6c", "problem_latex": "Discuss the system\n\\[\n\\begin{System}{3}\n x &+{}& y&+{}& z &= 1, \\\\\nax &+{}& by&+{}& cz &= k, \\\\\na^2x&+{}& b^2y&+{}& c^2z &= k^2,\n\\end{System}\n\\]\nwhen (\\emph{i})~$a$, $b$, $c$ are distinct; (\\emph{ii}) $a = b \\neq c$; (\\emph{iii}) $a = b = c$.", "markdown": "Discuss the system System3 x &+& y&+& z &= 1, ax &+& by&+& cz &= k, a^2x&+& b^2y&+& c^2z &= k^2, System when (*i*) $a$, $b$, $c$ are distinct; (*ii*) $a = b \\neq c$; (*iii*) $a = b = c$.", "answer_latex": [ "$z = 1 - x - y$ if $k=a$, inconsistent if $k\\ne a$." ], "answer_markdown": [ "$z = 1 - x - y$ if $k=a$, inconsistent if $k\\ne a$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page120/1", "set": "dickson-theory-of-equations-1922/ex-page120", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "120", "location": "Exercise Page120, problem 1", "problem_latex": "$\\begin{System}{3}\nx &+{}& y &+{}& 3z &= 0,\\\\\nx &+{}& 2y &+{}& 2z &= 0,\\\\\nx &+{}& 5y &-{}& z &= 0.\n\\end{System}$", "markdown": "$\\begin{System}{3} x &+{}& y &+{}& 3z &= 0,\\\\ x &+{}& 2y &+{}& 2z &= 0,\\\\ x &+{}& 5y &-{}& z &= 0. \\end{System}$", "answer_latex": [ "$r = 2$, $x:y:z = -4:1:1$." ], "answer_markdown": [ "$r = 2$, $x:y:z = -4:1:1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page120/2", "set": "dickson-theory-of-equations-1922/ex-page120", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "120", "location": "Exercise Page120, problem 2", "problem_latex": "$\\begin{System}{3}\n 2x &-{}& y &+{}& 4z &= 0,\\\\\n x &+{}& 3y &-{}& 2z &= 0,\\\\\n x &-{}& 11y &+{}& 14z &= 0.\n\\end{System}$", "markdown": "$\\begin{System}{3} 2x &-{}& y &+{}& 4z &= 0,\\\\ x &+{}& 3y &-{}& 2z &= 0,\\\\ x &-{}& 11y &+{}& 14z &= 0. \\end{System}$", "answer_latex": [ "$r = 2$, $x:y:z = -10:8:7$." ], "answer_markdown": [ "$r = 2$, $x:y:z = -10:8:7$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page120/3", "set": "dickson-theory-of-equations-1922/ex-page120", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "120", "location": "Exercise Page120, problem 3", "problem_latex": "$\\begin{System}{3}\n x &-{}& 3y &+{}& 4z &= 0,\\\\\n4x &-{}& 12y &+{}& 16z &= 0,\\\\\n3x &-{}& 9y &+{}& 12z &= 0.\n\\end{System}$", "markdown": "$\\begin{System}{3} x &-{}& 3y &+{}& 4z &= 0,\\\\ 4x &-{}& 12y &+{}& 16z &= 0,\\\\ 3x &-{}& 9y &+{}& 12z &= 0. \\end{System}$", "answer_latex": [ "$r = 1$, two unknowns arbitrary." ], "answer_markdown": [ "$r = 1$, two unknowns arbitrary." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page120/4", "set": "dickson-theory-of-equations-1922/ex-page120", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "120", "location": "Exercise Page120, problem 4", "problem_latex": "$\\begin{System}{4}\n6x &+{}& 4y &+{}& 3z &-{}& 84w &= 0,\\\\\n x &+{}& 2y &+{}& 3z &-{}& 48w &= 0,\\\\\n x &-{}& 2y &+{}& z &-{}& 12w &= 0,\\\\\n4x &+{}& 4y &-{}& z &-{}& 24w &= 0.\n\\end{System}$", "markdown": "$\\begin{System}{4} 6x &+{}& 4y &+{}& 3z &-{}& 84w &= 0,\\\\ x &+{}& 2y &+{}& 3z &-{}& 48w &= 0,\\\\ x &-{}& 2y &+{}& z &-{}& 12w &= 0,\\\\ 4x &+{}& 4y &-{}& z &-{}& 24w &= 0. \\end{System}$", "answer_latex": [ "$r = 3$, $x:y:z:w = 6:3:12:1$." ], "answer_markdown": [ "$r = 3$, $x:y:z:w = 6:3:12:1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page120/5", "set": "dickson-theory-of-equations-1922/ex-page120", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "120", "location": "Exercise Page120, problem 5", "problem_latex": "$\\begin{System}{4}\n2x &+{}& 3y &-{}& 4z &+{}& 5w &= 0,\\\\\n3x &+{}& 5y &-{}& z &+{}& 2w &= 0,\\\\\n7x &+{}& 11y &-{}& 9z &+{}& 12w &= 0,\\\\\n3x &+{}& 4y &-{}& 11z &+{}& 13w &= 0.\n\\end{System}$", "markdown": "$\\begin{System}{4} 2x &+{}& 3y &-{}& 4z &+{}& 5w &= 0,\\\\ 3x &+{}& 5y &-{}& z &+{}& 2w &= 0,\\\\ 7x &+{}& 11y &-{}& 9z &+{}& 12w &= 0,\\\\ 3x &+{}& 4y &-{}& 11z &+{}& 13w &= 0. \\end{System}$", "answer_latex": [ "$r = 2$, $z = -\\frac{11}{3} x - \\frac{19}{3} y$,\n $w = -\\frac{10}{3} x - \\frac{17}{3} y$." ], "answer_markdown": [ "$r = 2$, $z = -\\frac{11}{3} x - \\frac{19}{3} y$, $w = -\\frac{10}{3} x - \\frac{17}{3} y$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page121/1", "set": "dickson-theory-of-equations-1922/ex-page121", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Exercise Page121, problem 1", "problem_latex": "$\\begin{System}{3}\n 2x &+{}& y &+{}& 3z &= 1,\\\\\n 4x &+{}& 2y &-{}& z &= -3,\\\\\n 2x &+{}& y &-{}& 4z &= -4,\\\\\n10x &+{}& 5y &-{}& 6z &= -10.\n\\end{System}$", "markdown": "$\\begin{System}{3} 2x &+{}& y &+{}& 3z &= 1,\\\\ 4x &+{}& 2y &-{}& z &= -3,\\\\ 2x &+{}& y &-{}& 4z &= -4,\\\\ 10x &+{}& 5y &-{}& 6z &= -10. \\end{System}$", "answer_latex": [ "Ranks of $A$ and~$B$ are~$2$;\\quad $y = -8/7 - 2x, z = 5/7$." ], "answer_markdown": [ "Ranks of $A$ and $B$ are $2$; $y = -8/7 - 2x, z = 5/7$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page121/2", "set": "dickson-theory-of-equations-1922/ex-page121", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Exercise Page121, problem 2", "problem_latex": "$\\begin{System}{3}\n2x &-{}& y &+{}& 3z &= 2,\\\\\n x &+{}& 7y &+{}& z &= 1,\\\\\n3x &+{}& 5y &-{}& 5z &= a,\\\\\n4x &-{}& 3y &+{}& 2z &= 1.\n\\end{System}$", "markdown": "$\\begin{System}{3} 2x &-{}& y &+{}& 3z &= 2,\\\\ x &+{}& 7y &+{}& z &= 1,\\\\ 3x &+{}& 5y &-{}& 5z &= a,\\\\ 4x &-{}& 3y &+{}& 2z &= 1. \\end{System}$", "answer_latex": [ "Consistent only when $a = -225/61$ and then $x = -\\dfrac{5}{61}$, $y = \\dfrac{3}{61}$, $z = \\dfrac{45}{61}$." ], "answer_markdown": [ "Consistent only when $a = -225/61$ and then $x = -\\dfrac{5}{61}$, $y = \\dfrac{3}{61}$, $z = \\dfrac{45}{61}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": { "a": "-225/61", "x": "-5/61", "y": "3/61", "z": "45/61" } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(7*b + c + x, 1), Eq(-b + 3*c + 2*x, 2), Eq(5*b - 5*c + 3*x, a), Eq(-3*b + 2*c + 4*x, 1))" ], "shape": [ "solve: (Eq(N*b + c + x, 1), Eq(N*c + N*x - b, N), Eq(N*b + N*c + N*x, a), Eq(N*b + N*c + N*x, 1))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page121/3", "set": "dickson-theory-of-equations-1922/ex-page121", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Exercise Page121, problem 3", "problem_latex": "$\\begin{System}{3}\n4x &-{}& y &+{}& z &= 5,\\\\\n2x &-{}& 3y &+{}& 5z &= 1,\\\\\n x &+{}& y &-{}& 2z &= 2,\\\\\n5x & & &-{}& z &= 2.\n\\end{System}$", "markdown": "$\\begin{System}{3} 4x &-{}& y &+{}& z &= 5,\\\\ 2x &-{}& 3y &+{}& 5z &= 1,\\\\ x &+{}& y &-{}& 2z &= 2,\\\\ 5x & & &-{}& z &= 2. \\end{System}$", "answer_latex": [ "Rank of~$A$ is~$2$, rank of~$B$ is~$3$, inconsistent." ], "answer_markdown": [ "Rank of $A$ is $2$, rank of $B$ is $3$, inconsistent." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page121/4", "set": "dickson-theory-of-equations-1922/ex-page121", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Exercise Page121, problem 4", "problem_latex": "$\\begin{System}{2}\n 4x &-{}& 5y &= 2,\\\\\n 2x &+{}& 3y &= 12,\\\\\n10x &-{}& 7y &= 16.\n\\end{System}$", "markdown": "$\\begin{System}{2} 4x &-{}& 5y &= 2,\\\\ 2x &+{}& 3y &= 12,\\\\ 10x &-{}& 7y &= 16. \\end{System}$", "answer_latex": [ "$A$ and~$B$ of rank~$2$, $x = 3$, $y = 2$." ], "answer_markdown": [ "$A$ and $B$ of rank $2$, $x = 3$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": { "x": 3, "y": 2 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(3*a + 2*x, 12), Eq(-5*a + 4*x, 2), Eq(-7*a + 10*x, 16))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page121/5", "set": "dickson-theory-of-equations-1922/ex-page121", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Exercise Page121, problem 5", "problem_latex": "Prove the Corollary by multiplying the known terms by $x_{n+1}=1$ and applying~§97\nwith $n$ replaced by $n+1$.", "markdown": "Prove the Corollary by multiplying the known terms by $x_{n+1}=1$ and applying §97 with $n$ replaced by $n+1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page121/6", "set": "dickson-theory-of-equations-1922/ex-page121", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "121", "location": "Exercise Page121, problem 6", "problem_latex": "Prove that if the matrix of the coefficients of any system of linear homogeneous\n\\index{Linear equations!homogeneous}%\nequations in $n$~unknowns is of rank~$r$, the values of certain $n-r$ of the unknowns may be\n%% -----File: 128.png---Folio 122-------\nassigned at pleasure and the others will then be uniquely determined and satisfy all of the equations.", "markdown": "Prove that if the matrix of the coefficients of any system of linear homogeneous % equations in $n$ unknowns is of rank $r$, the values of certain $n-r$ of the unknowns may be %% -----File: 128.png---Folio 122------- assigned at pleasure and the others will then be uniquely determined and satisfy all of the equations.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:rank" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/1", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 1", "problem_latex": "Solve\n\\[\n\\begin{System}{3}\n ax &+{}& by &+{}& cz &= k,\\\\\na^2x &+{}& b^2y &+{}& c^2z &= k^2,\\\\\na^4x &+{}& b^4y &+{}& c^4z &= k^4\n\\end{System}\n\\]\nby determinants for~$x$, treating all cases.", "markdown": "Solve System3 ax &+& by &+& cz &= k, a^2x &+& b^2y &+& c^2z &= k^2, a^4x &+& b^4y &+& c^4z &= k^4 System by determinants for $x$, treating all cases.", "answer_latex": [ "$x = \\dfrac{k(b-k)(c-k)(k+b+c)}{a(b-a)(c-a)(a+b+c)}$, if $a$, $b$, $c$ are distinct and not zero and their\nsum $\\neq 0$. If $a = b \\neq c$, $ac \\ne 0$, equations are inconsistent unless $k = 0$, $a$, $c$, or $-a-c$,\nand then $y = \\dfrac{k(c-k)}{a(c-a)} - x$, $z = \\dfrac{k(k-a)}{c(c-a)}$, $x$~arbitrary." ], "answer_markdown": [ "$x = \\dfrac{k(b-k)(c-k)(k+b+c)}{a(b-a)(c-a)(a+b+c)}$, if $a$, $b$, $c$ are distinct and not zero and their sum $\\neq 0$. If $a = b \\neq c$, $ac \\ne 0$, equations are inconsistent unless $k = 0$, $a$, $c$, or $-a-c$, and then $y = \\dfrac{k(c-k)}{a(c-a)} - x$, $z = \\dfrac{k(k-a)}{c(c-a)}$, $x$ arbitrary." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(a*x + b*y + c*z, k) fails at {x: k*(b - k)*(c - k)*(b + c + k)/(a*(-a + b)*(-a + c)*(a + b + c))}: ('6.1043015389737523944', '0.0')", "problem_expr": "Eq(a*x + b*y + c*z, k)", "answer_expr": "k*(b-k)*(c-k)*(k+b+c)/(a*(b-a)*(c-a)*(a+b+c))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: (Eq(a*x + b*e + c*f, d), Eq(a**2*x + b**2*e + c**2*f, d**2), Eq(a**4*x + b**4*e + c**4*f, d**4))" ], "shape": [ "solve: (Eq(a*x + b*e + c*f, d), Eq(a**N*x + b**N*e + c**N*f, d**N), Eq(a**N*x + b**N*e + c**N*f, d**N))" ], "same_problem_in": [], "needs": [ "cas.factor", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/10", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 10", "problem_latex": "Prove that the cubic equation\n\\index{Cubic equation}%\n\\[\nD(x) \\equiv \\begin{vmatrix}\na-x & b & c \\\\\nb & f-x & g \\\\\nc & g & h-x\n\\end{vmatrix} = 0\n\\]\nhas only real roots. Hints:\n\\begin{gather*}\nD(x) · D(-x) = \\left|\\begin{array}{lll}\na^2+b^2+c^2-x^2 & ab+bf+cg & ac+bg+ch \\\\\nab+bf+cg & b^2+f^2+g^2-x^2 & bc+fg+gh \\\\\nac+bg+ch & bc+fg+gh & c^2+g^2+h^2-x^2\n\\end{array}\\right| \\\\\n%\n{} = -x^6+x^4(a^2+f^2+h^2+2b^2+2c^2+2g^2) - x^2(D_1+D_2+D_3)+ D^2(0),\n\\end{gather*}\nwhere $D_3$ denotes the first determinant in Ex.~9 with all accents removed and with\n$e = b$, while $D_1$ and~$D_2$ are analogous minors of elements in the main diagonal of the\npresent determinant of order~$3$ with $x = 0$. Hence the coefficient of~$-x^2$ is a sum of\nsquares. Since the function of degree~$6$ is not zero for a negative value of~$x^2$, $D(x)=0$\nhas no purely imaginary root. If it had an imaginary root $r+si$, then $D(x+r)=0$\nwould have a purely imaginary root~$si$. But $D(x+r)$ is of the form $D(x)$ with $a$, $f$, $h$\nreplaced by $a-r$, $f-r$, $h-r$. Hence $D(x)=0$ has only real roots. The method is\napplicable to such determinants of order~$n$.", "markdown": "Prove that the cubic equation % D(x) vmatrix a-x & b & c b & f-x & g c & g & h-x vmatrix = 0 has only real roots. Hints: gather* D(x) · D(-x) = |arraylll a^2+b^2+c^2-x^2 & ab+bf+cg & ac+bg+ch ab+bf+cg & b^2+f^2+g^2-x^2 & bc+fg+gh ac+bg+ch & bc+fg+gh & c^2+g^2+h^2-x^2 array| % = -x^6+x^4(a^2+f^2+h^2+2b^2+2c^2+2g^2) - x^2(D_1+D_2+D_3)+ D^2(0), gather* where $D_3$ denotes the first determinant in Ex. 9 with all accents removed and with $e = b$, while $D_1$ and $D_2$ are analogous minors of elements in the main diagonal of the present determinant of order $3$ with $x = 0$. Hence the coefficient of $-x^2$ is a sum of squares. Since the function of degree $6$ is not zero for a negative value of $x^2$, $D(x)=0$ has no purely imaginary root. If it had an imaginary root $r+si$, then $D(x+r)=0$ would have a purely imaginary root $si$. But $D(x+r)$ is of the form $D(x)$ with $a$, $f$, $h$ replaced by $a-r$, $f-r$, $h-r$. Hence $D(x)=0$ has only real roots. The method is applicable to such determinants of order $n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/11", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 11", "problem_latex": "If $a_1, \\dotsc, a_n$ are distinct, solve the system of equations\n\\[\n\\frac{x_1}{k_i-a_1} + \\frac{x_2}{k_i-a_2} + \\dotsb\n + \\frac{x_n}{k_i - a_n} = 1\\qquad (i=1, \\dotsc, n).\n\\]\n\nHint: Regard $k_1, \\dotsc, k_n$ as the roots of an equation of degree~$n$ in $k$ formed from\nthe typical one above by substituting~$k$ for~$k_i$ and clearing of fractions; write $k = a_j-t$,\nand consider the product of the roots of $t^n + \\dotsb = 0$. Hence find~$x_j$.", "markdown": "If $a_1, \\dotsc, a_n$ are distinct, solve the system of equations x_1k_i-a_1 + x_2k_i-a_2 + + x_nk_i - a_n = 1 (i=1, , n). Hint: Regard $k_1, \\dotsc, k_n$ as the roots of an equation of degree $n$ in $k$ formed from the typical one above by substituting $k$ for $k_i$ and clearing of fractions; write $k = a_j-t$, and consider the product of the roots of $t^n + \\dotsb = 0$. Hence find $x_j$.", "answer_latex": [ "$\\ds x_j = (k_1-a_j)\\dotsm(k_n-a_j)\n \\div \\prod\\limits^n_{\\substack{s=1 \\\\ s\\neq j}} (a_s-a_j)$." ], "answer_markdown": [ "$\\ds x_j = (k_1-a_j)\\dotsm(k_n-a_j) \\div \\prod\\limits^n_{\\substack{s=1 \\\\ s\\neq j}} (a_s-a_j)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/12", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 12", "problem_latex": "Solve the equation\n\\[\n\\begin{vmatrix}\na+x & x & x \\\\\n x & b+x & x \\\\\n x & x & c+x\n\\end{vmatrix} = 0.\n\\]", "markdown": "Solve the equation vmatrix a+x & x & x x & b+x & x x & x & c+x vmatrix = 0.", "answer_latex": [ "$x(ab + ac + bc) = -abc$." ], "answer_markdown": [ "$x(ab + ac + bc) = -abc$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(Matrix([[a+x, x, x], [x, b+x, x], [x, x, c+x]]).det(), 0)", "answer_expr": "-a*b*c/(a*b + a*c + b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(a*b*c + a*b*x + a*c*x + b*c*x, 0)" ], "shape": [ "solve: Eq(a*b*c + a*b*x + a*c*x + b*c*x, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/2", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 2", "problem_latex": "In three linear homogeneous equations in four unknowns, prove that the values\nof the unknowns are proportional to four determinants of order~$3$ formed from the\ncoefficients.", "markdown": "In three linear homogeneous equations in four unknowns, prove that the values of the unknowns are proportional to four determinants of order $3$ formed from the coefficients.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/3", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 3", "problem_latex": "$\\ds\n\\begin{vmatrix}\n1 & a & bc \\\\\n1 & b & ca \\\\\n1 & c & ab\n\\end{vmatrix}$.", "markdown": "$\\ds \\begin{vmatrix} 1 & a & bc \\\\ 1 & b & ca \\\\ 1 & c & ab \\end{vmatrix}$.", "answer_latex": [ "$(a-b)(b-c)(c-a)$." ], "answer_markdown": [ "$(a-b)(b-c)(c-a)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "Matrix([[1, a, b*c], [1, b, c*a], [1, c, a*b]]).det()", "answer_expr": "(a-b)*(b-c)*(c-a)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -a**2*b + a**2*c + a*b**2 - a*c**2 - b**2*c + b*c**2" ], "shape": [ "factor: a*b**N - a*c**N - a**N*b + a**N*c + b*c**N - b**N*c" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/4", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 4", "problem_latex": "$\\ds\n\\begin{vmatrix}\nx & x^2 & yz \\\\\ny & y^2 & xz \\\\\nz & z^2 & xy\n\\end{vmatrix} = \\begin{vmatrix}\nx^2 & x^3 & 1 \\\\\ny^2 & y^3 & 1 \\\\\nz^2 & z^3 & 1\n\\end{vmatrix}$.", "markdown": "$\\ds \\begin{vmatrix} x & x^2 & yz \\\\ y & y^2 & xz \\\\ z & z^2 & xy \\end{vmatrix} = \\begin{vmatrix} x^2 & x^3 & 1 \\\\ y^2 & y^3 & 1 \\\\ z^2 & z^3 & 1 \\end{vmatrix}$.", "answer_latex": [ "$(x-y)(y-z)(z-x)(xy + yz + zx)$." ], "answer_markdown": [ "$(x-y)(y-z)(z-x)(xy + yz + zx)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "Matrix([[x, x**2, y*z], [y, y**2, x*z], [z, z**2, x*y]]).det()", "answer_expr": "(x-y)*(y-z)*(z-x)*(x*y+y*z+z*x)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -a**3*b**2 + a**3*c**2 + a**2*b**3 - a**2*c**3 - b**3*c**2 + b**2*c**3" ], "shape": [ "factor: 0" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/5", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 5", "problem_latex": "\\[\n\\begin{vmatrix}\na & b & c \\\\\nc & a & b \\\\\nb & c & a\n\\end{vmatrix} = (a+b+c)(a+b\\omega+c\\omega^2)(a+b\\omega^2+c\\omega),\n\\]\nwhere $\\omega$ is an imaginary cube root of unity.", "markdown": "vmatrix a & b & c c & a & b b & c & a vmatrix = (a+b+c)(a+b+c^2)(a+b^2+c), where $\\omega$ is an imaginary cube root of unity.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "factor", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Matrix([[a, b, c], [c, a, b], [b, c, a]]).det()", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "factor: a**3 - 3*a*b*c + b**3 + c**3" ], "shape": [ "factor: N*a*b*c + a**N + b**N + c**N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/6", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 6", "problem_latex": "$\\ds\n\\begin{vmatrix}\na & b & c & d \\\\\nb & a & d & c \\\\\nc & d & a & b \\\\\nd & c & b & a\n\\end{vmatrix}$.", "markdown": "$\\ds \\begin{vmatrix} a & b & c & d \\\\ b & a & d & c \\\\ c & d & a & b \\\\ d & c & b & a \\end{vmatrix}$.", "answer_latex": [ "$(a+b+c+d)(a+b-c-d)(a-b-c+d)(a-b+c-d)$." ], "answer_markdown": [ "$(a+b+c+d)(a+b-c-d)(a-b-c+d)(a-b+c-d)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "Matrix([[a, b, c, d], [b, a, d, c], [c, d, a, b], [d, c, b, a]]).det()", "answer_expr": "(a+b+c+d)*(a+b-c-d)*(a-b-c+d)*(a-b+c-d)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: a**4 - 2*a**2*b**2 - 2*a**2*c**2 - 2*a**2*d**2 + 8*a*b*c*d + b**4 - 2*b**2*c**2 - 2*b**2*d**2 + c**4 - 2*c**2*d**2 + d**4" ], "shape": [ "factor: N*a*b*c*d + N*a**N*b**N + N*a**N*c**N + N*a**N*d**N + N*b**N*c**N + N*b**N*d**N + N*c**N*d**N + a**N + b**N + c**N + d**N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/7", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 7", "problem_latex": "$\\ds\n\\begin{vmatrix}\na & b & c & d \\\\\nd & a & b & c \\\\\nc & d & a & b \\\\\nb & c & d & a\n\\end{vmatrix}$.", "markdown": "$\\ds \\begin{vmatrix} a & b & c & d \\\\ d & a & b & c \\\\ c & d & a & b \\\\ b & c & d & a \\end{vmatrix}$.", "answer_latex": [ "$(a+b+c+d)(a-b+c-d)(a+bi-c-di)(a-bi-c+di)$." ], "answer_markdown": [ "$(a+b+c+d)(a-b+c-d)(a+bi-c-di)(a-bi-c+di)$." ], "checks": [ { "task": "factor", "verdict": "FLAG-MISMATCH", "judge_why": null, "problem_expr": "Matrix([[a, b, c, d], [d, a, b, c], [c, d, a, b], [b, c, d, a]]).det()", "answer_expr": "(a+b+c+d)*(a-b+c-d)*(a+b*I-c-d*I)*(a-b*I-c+d*I)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "factor: a**4 - 4*a**2*b*d - 2*a**2*c**2 + 4*a*b**2*c + 4*a*c*d**2 - b**4 + 2*b**2*d**2 - 4*b*c**2*d + c**4 - d**4" ], "shape": [ "factor: N*a*b**N*c + N*a*c*d**N + N*a**N*b*d + N*a**N*c**N + N*b*c**N*d + N*b**N*d**N + a**N - b**N + c**N - d**N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/8", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 8", "problem_latex": "If the points $(x_1, y_1), \\dotsc, (x_4, y_4)$ lie on a circle, prove that\n\\[\n\\left|\\begin{array}{cccc}\nx_1^2 + y_1^2 & x_1 & y_1 & 1 \\\\\n\\Dots{4} \\\\\nx_4^2 + y_4^2 & x_4 & y_4 & 1\n\\end{array}\\right| = 0.\n\\]", "markdown": "If the points $(x_1, y_1), \\dotsc, (x_4, y_4)$ lie on a circle, prove that |arraycccc x_1^2 + y_1^2 & x_1 & y_1 & 1 [2]4 x_4^2 + y_4^2 & x_4 & y_4 & 1 array| = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page126/9", "set": "dickson-theory-of-equations-1922/ex-page126", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "126", "location": "Exercise Page126, problem 9", "problem_latex": "Prove that\n\\begin{gather*}\n\\begin{vmatrix}\naa' + bb' + cc' & ea' + fb' + gc' \\\\\nae' + bf' + cg' & ee' + ff' + gg'\n\\end{vmatrix} \\\\\n%\n{} = \\begin{vmatrix}\na & b \\\\\ne & f\n\\end{vmatrix} · \\begin{vmatrix}\na' & b' \\\\\ne' & f'\n\\end{vmatrix} + \\begin{vmatrix}\na & c \\\\\ne & g\n\\end{vmatrix} · \\begin{vmatrix}\na' & c' \\\\\ne' & g'\n\\end{vmatrix} + \\begin{vmatrix}\nb & c \\\\\nf & g\n\\end{vmatrix} · \\begin{vmatrix}\nb' & c' \\\\\nf' & g'\n\\end{vmatrix}.\n\\end{gather*}", "markdown": "Prove that gather* vmatrix aa’ + bb’ + cc’ & ea’ + fb’ + gc’ ae’ + bf’ + cg’ & ee’ + ff’ + gg’ vmatrix % = vmatrix a & b e & f vmatrix · vmatrix a’ & b’ e’ & f’ vmatrix + vmatrix a & c e & g vmatrix · vmatrix a’ & c’ e’ & g’ vmatrix + vmatrix b & c f & g vmatrix · vmatrix b’ & c’ f’ & g’ vmatrix. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/1", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 1", "problem_latex": "$x^4 - 3x^2 - x - 6$ is divided by $x + 3$.", "markdown": "$x^4 - 3x^2 - x - 6$ is divided by $x + 3$.", "answer_latex": [ "$51$." ], "answer_markdown": [ "$51$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 51.0, printed 51 (half-unit 0.5; correctly rounded at the printed digits: 51.0)", "problem_expr": "x**4 - 3*x**2 - x - 6", "answer_expr": "51" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: x**4 - 3*x**2 - x - 6 at x=-3" ], "shape": [ "evaluate: N*x**N + N - x + x**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#51,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 +/− GOLD x²", "ENTER GOLD x²", "x↔y 3 ×", "− 3 +", "6 −" ], "constants": [ { "value": "3", "source": "the 3 in '-3x^2' in the problem's expression (coefficient of x²)" }, { "value": "3", "source": "the given value x = -3 (its magnitude, typed as 3 then +/−, and again as 3 for the '-x' term, which becomes +3 when x = -3)" }, { "value": "6", "source": "the 6 in '-6' in the problem's expression (constant term)" } ], "calculator_value": "+51E+0", "printed_value": "51", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page13/10", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 10", "problem_latex": "If $a$, $ar$, $ar^2, \\dotsc, ar^{n-1}$ are $n$ numbers in \\emph{geometrical progression} (the ratio of any\nterm to the preceding being a constant $r \\ne 1$), prove by Exercise~7 that their sum is\nequal to\n\\index{Geometrical!progression}%\n\\[\n\\frac{a(r^n - 1)}{r - 1}.\n\\]", "markdown": "If $a$, $ar$, $ar^2, \\dotsc, ar^{n-1}$ are $n$ numbers in *geometrical progression* (the ratio of any term to the preceding being a constant $r \\ne 1$), prove by Exercise 7 that their sum is equal to % a(r^n - 1)r - 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/11", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 11", "problem_latex": "At the end of each of $n$ years a man deposits in a savings bank $a$~dollars. With\nannual compound interest at~4\\%, show that his account at the end of $n$~years will be\n\\index{Compound interest}%\n\\[\n\\frac{a}{.04} \\bigl\\{(1.04)^n - 1\\bigr\\}\n\\]\ndollars. Hint: The final deposit draws no interest; the prior deposit will amount to\n$a(1.04)$ dollars; the deposit preceding that will amount to $a(1.04)^2$ dollars, etc. Hence\napply Exercise~10 for $r = 1.04$.", "markdown": "At the end of each of $n$ years a man deposits in a savings bank $a$ dollars. With annual compound interest at 4%, show that his account at the end of $n$ years will be % a.04 (1.04)^n - 1 dollars. Hint: The final deposit draws no interest; the prior deposit will amount to $a(1.04)$ dollars; the deposit preceding that will amount to $a(1.04)^2$ dollars, etc. Hence apply Exercise 10 for $r = 1.04$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/2", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 2", "problem_latex": "$x^3 - 3x^2 + 6x - 5$ is divided by $x - 3$.", "markdown": "$x^3 - 3x^2 + 6x - 5$ is divided by $x - 3$.", "answer_latex": [ "$13$." ], "answer_markdown": [ "$13$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 13.0, printed 13 (half-unit 0.5; correctly rounded at the printed digits: 13.0)", "problem_expr": "x**3 - 3*x**2 + 6*x - 5", "answer_expr": "13" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: x**3 - 3*x**2 + 6*x - 5 at x=3" ], "shape": [ "evaluate: N*x + N*x**N + N + x**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#13,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 3 yˣ", "3 ENTER GOLD x² ×", "−", "6 ENTER 3 × +", "5 −" ], "calculator_value": "+1300000000000000000000000000000000E-32", "printed_value": "13", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page13/3", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 3", "problem_latex": "$18x^{10} + 19x^5 + 1$ is divisible by $x + 1$.", "markdown": "$18x^{10} + 19x^5 + 1$ is divisible by $x + 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/4", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 4", "problem_latex": "$2x^4 - x^3 - 6x^2 + 4x - 8$ is divisible by $x - 2$ and $x + 2$.", "markdown": "$2x^4 - x^3 - 6x^2 + 4x - 8$ is divisible by $x - 2$ and $x + 2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/5", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 5", "problem_latex": "$x^4 - 3x^3 + 3x^2 - 3x + 2$ is divisible by $x - 1$ and $x - 2$.", "markdown": "$x^4 - 3x^3 + 3x^2 - 3x + 2$ is divisible by $x - 1$ and $x - 2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/6", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 6", "problem_latex": "$r^3 - 1$, $r^4 - 1$, $r^5 - 1$ are divisible by $r - 1$.", "markdown": "$r^3 - 1$, $r^4 - 1$, $r^5 - 1$ are divisible by $r - 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/7", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 7", "problem_latex": "By performing the indicated multiplication, verify that\n\\[\nr^n - 1 \\equiv (r - 1)(r^{n-1} + r^{n-2} + \\dotsb + r + 1).\n\\]", "markdown": "By performing the indicated multiplication, verify that r^n - 1 (r - 1)(r^n-1 + r^n-2 + + r + 1).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/8", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 8", "problem_latex": "In the last identity replace $r$ by~$x/y$, multiply by~$y^n$, and derive\n\\[\nx^n - y^n \\equiv (x-y)(x^{n-1} + x^{n-2}y + \\dotsb + xy^{n-2} + y^{n-1}).\n\\]", "markdown": "In the last identity replace $r$ by $x/y$, multiply by $y^n$, and derive x^n - y^n (x-y)(x^n-1 + x^n-2y + + xy^n-2 + y^n-1).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page13/9", "set": "dickson-theory-of-equations-1922/ex-page13", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "13", "location": "Exercise Page13, problem 9", "problem_latex": "In the identity of Exercise~8 replace $y$ by $-y$, and derive\n\\begin{align*}\nx^n + y^n\n &\\equiv (x+y)(x^{n-1} - x^{n-2} y + \\dotsb - xy^{n-2} + y^{n-1}),\n \\quad \\text{$n$~odd}; \\\\\nx^n - y^n\n &\\equiv (x+y)(x^{n-1} - x^{n-2} y + \\dotsb + xy^{n-2} - y^{n-1}),\n \\quad\\text{$n$~even}.\n\\end{align*}", "markdown": "In the identity of Exercise 8 replace $y$ by $-y$, and derive align* x^n + y^n &(x+y)(x^n-1 - x^n-2 y + - xy^n-2 + y^n-1), $n$ odd; x^n - y^n &(x+y)(x^n-1 - x^n-2 y + + xy^n-2 - y^n-1), $n$ even. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/1", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 1", "problem_latex": "$\\Sigma \\dfrac{\\beta\\gamma + \\alpha^2}{\\beta + \\gamma}$, % [** PP: Added ,]", "markdown": "$\\Sigma \\dfrac{\\beta\\gamma + \\alpha^2}{\\beta + \\gamma}$, % [** PP: Added ,]", "answer_latex": [ "$\\dfrac{p^4 - 3p^2q + 5pr + q^2}{r - pq}$." ], "answer_markdown": [ "$\\dfrac{p^4 - 3p^2q + 5pr + q^2}{r - pq}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(beta*gamma + alpha**2)/(beta + gamma) + (gamma*alpha + beta**2)/(gamma + alpha) + (alpha*beta + gamma**2)/(alpha + beta)", "answer_expr": "(p**4 - 3*p**2*q + 5*p*r + q**2)/(r - p*q)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/10", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 10", "problem_latex": "$\\alpha\\beta + \\alpha\\gamma$,\n$\\alpha\\beta + \\beta\\gamma$,\n$\\alpha\\gamma + \\beta\\gamma$.", "markdown": "$\\alpha\\beta + \\alpha\\gamma$, $\\alpha\\beta + \\beta\\gamma$, $\\alpha\\gamma + \\beta\\gamma$.", "answer_latex": [ "$y = q+r/x$." ], "answer_markdown": [ "$y = q+r/x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "Eq(y, q + r/x)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/11", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 11", "problem_latex": "$\\dfrac{2\\alpha - 1}{\\beta + \\gamma - \\alpha}$, etc.", "markdown": "$\\dfrac{2\\alpha - 1}{\\beta + \\gamma - \\alpha}$, etc.", "answer_latex": [ "$x = \\dfrac{1-py}{2+2y}$." ], "answer_markdown": [ "$x = \\dfrac{1-py}{2+2y}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(2*alpha - 1)/(beta + gamma - alpha)", "answer_expr": "Eq(x, (1 - p*y)/(2 + 2*y))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/12", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 12", "problem_latex": "$\\dfrac{\\beta\\gamma + 3\\alpha^2}{\\beta + \\gamma - 2\\alpha}$, etc.", "markdown": "$\\dfrac{\\beta\\gamma + 3\\alpha^2}{\\beta + \\gamma - 2\\alpha}$, etc.", "answer_latex": [ "$y = \\dfrac{4x^2 + px + q}{-3x-p}$, see~§112." ], "answer_markdown": [ "$y = \\dfrac{4x^2 + px + q}{-3x-p}$, see §112." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(beta*gamma + 3*alpha**2)/(beta + gamma - 2*alpha)", "answer_expr": "Eq(y, (4*x**2 + p*x + q)/(-3*x - p))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/13", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 13, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 13", "problem_latex": "$\\Sigma\\dfrac{\\beta^2 + \\gamma^2 + \\delta^2}{\\beta + \\gamma + \\delta}$.", "markdown": "$\\Sigma\\dfrac{\\beta^2 + \\gamma^2 + \\delta^2}{\\beta + \\gamma + \\delta}$.", "answer_latex": [ "$\\dfrac{2q(p^3 + 2pq - r)}{p^2q - pr + s} - 5p$, see Ex.~17." ], "answer_markdown": [ "$\\dfrac{2q(p^3 + 2pq - r)}{p^2q - pr + s} - 5p$, see Ex. 17." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(beta**2 + gamma**2 + delta**2)/(beta + gamma + delta) + (alpha**2 + gamma**2 + delta**2)/(alpha + gamma + delta) + (alpha**2 + beta**2 + delta**2)/(alpha + beta + delta) + (alpha**2 + beta**2 + gamma**2)/(alpha + beta + gamma)", "answer_expr": "2*q*(p**3 + 2*p*q - r)/(p**2*q - p*r + s) - 5*p" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/14", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 14", "problem_latex": "$\\Sigma\\dfrac{\\beta\\gamma + \\beta\\delta + \\gamma\\delta}{\\beta + \\gamma + \\delta - 3}$.", "markdown": "$\\Sigma\\dfrac{\\beta\\gamma + \\beta\\delta + \\gamma\\delta}{\\beta + \\gamma + \\delta - 3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(beta*gamma + beta*delta + gamma*delta)/(beta + gamma + delta - 3) + (alpha*gamma + alpha*delta + gamma*delta)/(alpha + gamma + delta - 3) + (alpha*beta + alpha*delta + beta*delta)/(alpha + beta + delta - 3) + (alpha*beta + alpha*gamma + beta*gamma)/(alpha + beta + gamma - 3)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/15", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 15, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 15", "problem_latex": "Prove that if $y_1$, $y_2$, $y_3$ are the roots of $y^3 + py + q = 0$, the equation with the roots\n$z_1 = (y_2 - y_3)^2$, $z_2 = (y_1 - y_3)^2$, $z_3 = (y_1 - y_2)^2$ is\n\\index{Equation for differences of roots!squares of differences}%\n\\[\nz^3 + 6pz^2 + 9p^2 z + 4p^3 + 27q^2 = 0.\n\\]\nHints: since $z_1 = \\Sigma y_1^2 - 2y_2y_3 - y_1^2 = -2p + 2q/y_1 - y_1^2$, etc., we set $z = -2p + 2q/y - y^2$.\nBy the given equation, $y^2 + p + q/y = 0$. Thus the desired substitution is $z = -p + 3q/y$,\n$y = 3q/(z + p)$.", "markdown": "Prove that if $y_1$, $y_2$, $y_3$ are the roots of $y^3 + py + q = 0$, the equation with the roots $z_1 = (y_2 - y_3)^2$, $z_2 = (y_1 - y_3)^2$, $z_3 = (y_1 - y_2)^2$ is % z^3 + 6pz^2 + 9p^2 z + 4p^3 + 27q^2 = 0. Hints: since $z_1 = \\Sigma y_1^2 - 2y_2y_3 - y_1^2 = -2p + 2q/y_1 - y_1^2$, etc., we set $z = -2p + 2q/y - y^2$. By the given equation, $y^2 + p + q/y = 0$. Thus the desired substitution is $z = -p + 3q/y$, $y = 3q/(z + p)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/16", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 16, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 16", "problem_latex": "Hence find the discriminant of the reduced cubic equation.\n\\index{Cubic equation|)}%\n\\index{Discriminant!of cubic}%", "markdown": "Hence find the discriminant of the reduced cubic equation. % %", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/17", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 17, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 17", "problem_latex": "If $x_1, \\dotsc, x_n$ are the roots of $f(x)=0$, show that\n\\[\n\\Sigma \\frac{1}{x_1 - c} = \\frac{-f'(c)}{f(c)}.\n\\]\nHint: $x_1 - c = y_1, \\dotsc, x_n - c = y_n$ are the roots of\n\\[\nf(c+y) = f(c) + yf'(c) + y^2(\\quad)+ \\dotsb = 0,\n\\]\nas shown by Taylor's theorem. Or we may employ~\\Eq{5} below % [** PP: Added `below']\nfor $x = c$.", "markdown": "If $x_1, \\dotsc, x_n$ are the roots of $f(x)=0$, show that 1x_1 - c = -f’(c)f(c). Hint: $x_1 - c = y_1, \\dotsc, x_n - c = y_n$ are the roots of f(c+y) = f(c) + yf’(c) + y^2( )+ = 0, as shown by Taylor’s theorem. Or we may employ $(5)$ below % [** PP: Added ‘below’] for $x = c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/2", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 2", "problem_latex": "$\\Sigma \\dfrac{3\\beta\\gamma - 2\\alpha^2}{\\beta + \\gamma - \\alpha}$.", "markdown": "$\\Sigma \\dfrac{3\\beta\\gamma - 2\\alpha^2}{\\beta + \\gamma - \\alpha}$.", "answer_latex": [ "$\\dfrac{(5p^2-12q)(p^2-4q)}{4(p^3 - 4pq + 8r)} - \\dfrac{13}{4}p$." ], "answer_markdown": [ "$\\dfrac{(5p^2-12q)(p^2-4q)}{4(p^3 - 4pq + 8r)} - \\dfrac{13}{4}p$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(3*beta*gamma - 2*alpha**2)/(beta + gamma - alpha) + (3*gamma*alpha - 2*beta**2)/(gamma + alpha - beta) + (3*alpha*beta - 2*gamma**2)/(alpha + beta - gamma)", "answer_expr": "(5*p**2 - 12*q)*(p**2 - 4*q)/(4*(p**3 - 4*p*q + 8*r)) - 13*p/4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/3", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 3", "problem_latex": "Why would the use of $\\beta\\gamma = -r/\\alpha$ complicate Exs.\\ 1,~2? Verify that\n\\[\n\\beta\\gamma\n = \\frac{-r}{\\alpha}\n = \\frac{f(\\alpha) - r}{\\alpha}\n = \\alpha^2 + p \\alpha + q.\n\\]", "markdown": "Why would the use of $\\beta\\gamma = -r/\\alpha$ complicate Exs. 1, 2? Verify that = -r = f() - r = ^2 + p + q.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/4", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 4", "problem_latex": "Why would you use $\\beta\\gamma = -r/\\alpha$ in finding\n $\\Sigma \\dfrac{\\beta^2 + \\gamma^2}{\\beta\\gamma + c}$?", "markdown": "Why would you use $\\beta\\gamma = -r/\\alpha$ in finding $\\Sigma \\dfrac{\\beta^2 + \\gamma^2}{\\beta\\gamma + c}$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/5", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 5", "problem_latex": "Find $\\Sigma (\\beta + \\gamma)^2$.", "markdown": "Find $\\Sigma (\\beta + \\gamma)^2$.", "answer_latex": [ "$2p^2-2q$." ], "answer_markdown": [ "$2p^2-2q$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(beta + gamma)**2 + (gamma + alpha)**2 + (alpha + beta)**2", "answer_expr": "2*p**2 - 2*q" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/6", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 6", "problem_latex": "Find $\\Sigma (\\alpha + \\beta - \\gamma)^3$.", "markdown": "Find $\\Sigma (\\alpha + \\beta - \\gamma)^3$.", "answer_latex": [ "$24r-p^3$." ], "answer_markdown": [ "$24r-p^3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(alpha + beta - gamma)**3 + (beta + gamma - alpha)**3 + (gamma + alpha - beta)**3", "answer_expr": "24*r - p**3" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/7", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 7", "problem_latex": "Find $\\smash{\\Sigma \\left(\\dfrac{\\beta - \\gamma}{\\beta + \\gamma}\\right)^2}$.", "markdown": "Find $\\smash{\\Sigma \\left(\\dfrac{\\beta - \\gamma}{\\beta + \\gamma}\\right)^2}$.", "answer_latex": [ "$\\dfrac{3p^2q^2 - 4p^3r - 4q^3 - 2pqr - 9r^2}{(r - pq)^2}$." ], "answer_markdown": [ "$\\dfrac{3p^2q^2 - 4p^3r - 4q^3 - 2pqr - 9r^2}{(r - pq)^2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "((beta - gamma)/(beta + gamma))**2 + ((gamma - alpha)/(gamma + alpha))**2 + ((alpha - beta)/(alpha + beta))**2", "answer_expr": "(3*p**2*q**2 - 4*p**3*r - 4*q**3 - 2*p*q*r - 9*r**2)/(r - p*q)**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/8", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 8", "problem_latex": "Find a necessary and sufficient condition on the coefficients that the roots, in\nsome order, shall be in harmonic progression.\nHint: If $\\dfrac{1}{\\alpha} + \\dfrac{1}{\\gamma} = \\dfrac{2}{\\beta}$, then $\\dfrac{-3r}{q} - \\beta = 0$,\nand conversely. Hence the condition is\n\\[\n\\left(\\frac{-3r}{q} - \\alpha\\right)\n\\left(\\frac{-3r}{q} - \\beta\\right)\n\\left(\\frac{-3r}{q} - \\gamma\\right)\n = f\\left(\\frac{-3r}{q}\\right)\n = 0.\n\\]", "markdown": "Find a necessary and sufficient condition on the coefficients that the roots, in some order, shall be in harmonic progression. Hint: If $\\dfrac{1}{\\alpha} + \\dfrac{1}{\\gamma} = \\dfrac{2}{\\beta}$, then $\\dfrac{-3r}{q} - \\beta = 0$, and conversely. Hence the condition is (-3rq - ) (-3rq - ) (-3rq - ) = f(-3rq) = 0.", "answer_latex": [ "$27r^2 - 9pqr + 2q^3 = 0$." ], "answer_markdown": [ "$27r^2 - 9pqr + 2q^3 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "Eq(27*r**2 - 9*p*q*r + 2*q**3, 0)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page133/9", "set": "dickson-theory-of-equations-1922/ex-page133", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "133", "location": "Exercise Page133, problem 9", "problem_latex": "Find the cubic equation with the roots\n$\\beta\\gamma - \\dfrac{1}{\\alpha}$,\n$\\alpha\\gamma - \\dfrac{1}{\\beta}$,\n$\\alpha\\beta - \\dfrac{1}{\\gamma}$.\nHint: since these are $(-r - 1)/\\alpha$, etc., make the substitution $(-r - 1)/x = y$.", "markdown": "Find the cubic equation with the roots $\\beta\\gamma - \\dfrac{1}{\\alpha}$, $\\alpha\\gamma - \\dfrac{1}{\\beta}$, $\\alpha\\beta - \\dfrac{1}{\\gamma}$. Hint: since these are $(-r - 1)/\\alpha$, etc., make the substitution $(-r - 1)/x = y$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/1", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 1", "problem_latex": "For a cubic equation, $s_4 = c_1^4 - 4c_1^2 c_2 + 4c_1 c_3 + 2c_2^2$.", "markdown": "For a cubic equation, $s_4 = c_1^4 - 4c_1^2 c_2 + 4c_1 c_3 + 2c_2^2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/2", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 2", "problem_latex": "For an equation of degree $\\geqq 4$, $s_4 = c_1^4 - 4c_1^2 c_2 + 4c_1 c_3 + 2c_2^2- 4c_4$.", "markdown": "For an equation of degree $\\geqq 4$, $s_4 = c_1^4 - 4c_1^2 c_2 + 4c_1 c_3 + 2c_2^2- 4c_4$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/3a", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 3, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 3a", "problem_latex": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "answer_latex": [ "$s_2 = p^2 - 2q$, \\\\" ], "answer_markdown": [ "$s_2 = p^2 - 2q$," ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**2 - p*x + q", "answer_expr": "p**2 - 2*q" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/3b", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 3, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 3b", "problem_latex": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "answer_latex": [ "$s_3 = p^3 - 3pq$, \\\\" ], "answer_markdown": [ "$s_3 = p^3 - 3pq$," ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**2 - p*x + q", "answer_expr": "p**3 - 3*p*q" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/3c", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 3, "part": "c", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 3c", "problem_latex": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "answer_latex": [ "$s_4 = p^4 - 4p^2q + 2q^2$, \\\\" ], "answer_markdown": [ "$s_4 = p^4 - 4p^2q + 2q^2$," ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**2 - p*x + q", "answer_expr": "p**4 - 4*p**2*q + 2*q**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/3d", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 3, "part": "d", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 3d", "problem_latex": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_4$, $s_5$ for $x^2 - px + q = 0$.", "answer_latex": [ "$s_5 = p^5 - 5p^3q + 5pq^2$." ], "answer_markdown": [ "$s_5 = p^5 - 5p^3q + 5pq^2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**2 - p*x + q", "answer_expr": "p**5 - 5*p**3*q + 5*p*q**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/4", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 4", "problem_latex": "Find $s_k$ for $x^5 - 3 = 0$.", "markdown": "Find $s_k$ for $x^5 - 3 = 0$.", "answer_latex": [ "$s_{5n} = 5·3^n$, \\\\\n $s_k = 0$ if $k$ is not divisible by~$5$." ], "answer_markdown": [ "$s_{5n} = 5·3^n$, $s_k = 0$ if $k$ is not divisible by $5$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**5 - 3", "answer_expr": "Piecewise((5*3**(k/5), Mod(k, 5) == 0), (0, True))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/5a", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 5, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 5a", "problem_latex": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "answer_latex": [ "All zero." ], "answer_markdown": [ "All zero." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**5 - p*x + q", "answer_expr": "0" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/5b", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 5, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 5b", "problem_latex": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "answer_latex": [ "All zero." ], "answer_markdown": [ "All zero." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**5 - p*x + q", "answer_expr": "0" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/5c", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 5, "part": "c", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 5c", "problem_latex": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "answer_latex": [ "All zero." ], "answer_markdown": [ "All zero." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**5 - p*x + q", "answer_expr": "0" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page136/5d", "set": "dickson-theory-of-equations-1922/ex-page136", "number": 5, "part": "d", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "136", "location": "Exercise Page136, problem 5d", "problem_latex": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "markdown": "Find $s_2$, $s_3$, $s_6$, $s_7$ for $x^5 - px + q = 0$.", "answer_latex": [ "All zero." ], "answer_markdown": [ "All zero." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**5 - p*x + q", "answer_expr": "0" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page140/1", "set": "dickson-theory-of-equations-1922/ex-page140", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "140", "location": "Exercise Page140, problem 1", "problem_latex": "For the quadratic $x^2 - px + q = 0$ write out the expressions for $s_2$, $s_3$, $s_4$, $s_5$ given by~\\Eq{19},\nand compare with those obtained from Newton's identities (Ex.~3, §106).", "markdown": "For the quadratic $x^2 - px + q = 0$ write out the expressions for $s_2$, $s_3$, $s_4$, $s_5$ given by $(19)$, and compare with those obtained from Newton’s identities (Ex. 3, §106).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page140/2", "set": "dickson-theory-of-equations-1922/ex-page140", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "140", "location": "Exercise Page140, problem 2", "problem_latex": "Find $s_4$ for a quartic equation by Waring's formula.", "markdown": "Find $s_4$ for a quartic equation by Waring’s formula.", "answer_latex": [ "See Ex.~2, p.~136." ], "answer_markdown": [ "See Ex. 2, p. 136." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page140/3", "set": "dickson-theory-of-equations-1922/ex-page140", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "140", "location": "Exercise Page140, problem 3", "problem_latex": "For $k=5$, \\Eq{20} becomes De Moivre's quintic $p^5 - 5qp^3 + 5q^2p = c$. Solve it by\nradicals for~$p$.", "markdown": "For $k=5$, $(20)$ becomes De Moivre’s quintic $p^5 - 5qp^3 + 5q^2p = c$. Solve it by radicals for $p$.", "answer_latex": [ "$\\epsilon^j \\sqrt[5]{\\frac{1}{2}c + \\sqrt{Q}}\n + \\epsilon^{5-j} \\sqrt[5]{\\frac{1}{2}c - \\sqrt{Q}}$,\\quad\n $Q = \\frac{1}{4}c^2 - q^5$\\hfill ($j=0$, $1$, $2$, $3$, $4$)." ], "answer_markdown": [ "$\\epsilon^j \\sqrt[5]{\\frac{1}{2}c + \\sqrt{Q}} + \\epsilon^{5-j} \\sqrt[5]{\\frac{1}{2}c - \\sqrt{Q}}$, $Q = \\frac{1}{4}c^2 - q^5$($j=0$, $1$, $2$, $3$, $4$)." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(p**5 - 5*p**3*q + 5*p*q**2, c) fails at {p: epsilon**j*(c/2 + sqrt(c**2/4 - q**5))**(1/5) + epsilon**(5 - j)*(c/2 - sqrt(c**2/4 - q**5))**(1/5)}: None", "problem_expr": "Eq(p**5 - 5*q*p**3 + 5*q**2*p, c)", "answer_expr": "epsilon**j*(c/2 + sqrt(c**2/4 - q**5))**Rational(1,5) + epsilon**(5-j)*(c/2 - sqrt(c**2/4 - q**5))**Rational(1,5)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(5*b**2*x - 5*b*x**3 + x**5, a)" ], "shape": [ "solve: Eq(N*b*x**N + N*b**N*x + x**N, a)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page140/4", "set": "dickson-theory-of-equations-1922/ex-page140", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "140", "location": "Exercise Page140, problem 4", "problem_latex": "Solve \\Eq{20} by radicals when $k=7$.", "markdown": "Solve $(20)$ by radicals when $k=7$.", "answer_latex": [ "$\\epsilon^j \\sqrt[7]{\\frac{1}{2}c + \\sqrt{Q}}\n + \\epsilon^{7-j} \\sqrt[7]{\\frac{1}{2}c - \\sqrt{Q}}$,\\quad\n $Q = \\frac{1}{4}c^2 - q^7$\\hfill ($j=0$, $1,\\dotsc, 6$)." ], "answer_markdown": [ "$\\epsilon^j \\sqrt[7]{\\frac{1}{2}c + \\sqrt{Q}} + \\epsilon^{7-j} \\sqrt[7]{\\frac{1}{2}c - \\sqrt{Q}}$, $Q = \\frac{1}{4}c^2 - q^7$($j=0$, $1,\\dotsc, 6$)." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": "epsilon**j*(c/2 + sqrt(c**2/4 - q**7))**Rational(1,7) + epsilon**(7-j)*(c/2 - sqrt(c**2/4 - q**7))**Rational(1,7)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141/1", "set": "dickson-theory-of-equations-1922/ex-page141", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "141", "location": "Exercise Page141, problem 1", "problem_latex": "$\\Sigma \\alpha_1^2 \\alpha_2^2$.", "markdown": "$\\Sigma \\alpha_1^2 \\alpha_2^2$.", "answer_latex": [ "$c_2^2 - 2c_1c_3 + 2c_4$." ], "answer_markdown": [ "$c_2^2 - 2c_1c_3 + 2c_4$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c_2**2 - 2*c_1*c_3 + 2*c_4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:symmetric_function_reduction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141/2", "set": "dickson-theory-of-equations-1922/ex-page141", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "141", "location": "Exercise Page141, problem 2", "problem_latex": "$\\Sigma \\alpha_1^3 \\alpha_2$.", "markdown": "$\\Sigma \\alpha_1^3 \\alpha_2$.", "answer_latex": [ "$c_1^2c_2 - 2c_2^2 - c_1c_3 + 4c_4$." ], "answer_markdown": [ "$c_1^2c_2 - 2c_2^2 - c_1c_3 + 4c_4$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c_1**2*c_2 - 2*c_2**2 - c_1*c_3 + 4*c_4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:symmetric_function_reduction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141/3", "set": "dickson-theory-of-equations-1922/ex-page141", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "141", "location": "Exercise Page141, problem 3", "problem_latex": "$\\Sigma \\alpha_1^2 \\alpha_2 \\alpha_3$.", "markdown": "$\\Sigma \\alpha_1^2 \\alpha_2 \\alpha_3$.", "answer_latex": [ "$c_1c_3 - 4c_4$." ], "answer_markdown": [ "$c_1c_3 - 4c_4$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c_1*c_3 - 4*c_4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:symmetric_function_reduction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141/4", "set": "dickson-theory-of-equations-1922/ex-page141", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "141", "location": "Exercise Page141, problem 4", "problem_latex": "$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3^2$.", "markdown": "$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3^2$.", "answer_latex": [ "$c_3^2 - 2c_2c_4$." ], "answer_markdown": [ "$c_3^2 - 2c_2c_4$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c_3**2 - 2*c_2*c_4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:symmetric_function_reduction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141/5", "set": "dickson-theory-of-equations-1922/ex-page141", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "141", "location": "Exercise Page141, problem 5", "problem_latex": "If $a\\geqq b > c > 0$, prove that\n\\[\n\\Sigma \\alpha_1^a \\alpha_2^b \\alpha_3^c\n = \\frac{1}{m} (s_a s_b s_c - s_a s_{b+c}\n - s_b s_{a+c} - s_c s_{a+b} + 2s_{a+b+c}),\n\\]\nwhere $m = 1$ if $a > b$, $m = 2$ if $a = b$.", "markdown": "If $a\\geqq b > c > 0$, prove that _1^a _2^b _3^c = 1m (s_a s_b s_c - s_a s_b+c - s_b s_a+c - s_c s_a+b + 2s_a+b+c), where $m = 1$ if $a > b$, $m = 2$ if $a = b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:symmetric_function_identity_proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141/6", "set": "dickson-theory-of-equations-1922/ex-page141", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "141", "location": "Exercise Page141, problem 6", "problem_latex": "$\\Sigma \\alpha_1^a \\alpha_2^b \\alpha_3^b\n = \\frac{1}{2}(s_a s_b^2 - s_as_{2b} - 2s_b s_{a+b} + 2s_{a+2b})$,\\qquad $a > b > 0$.", "markdown": "$\\Sigma \\alpha_1^a \\alpha_2^b \\alpha_3^b = \\frac{1}{2}(s_a s_b^2 - s_as_{2b} - 2s_b s_{a+b} + 2s_{a+2b})$, $a > b > 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:symmetric_function_identity_proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page141/7", "set": "dickson-theory-of-equations-1922/ex-page141", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "141", "location": "Exercise Page141, problem 7", "problem_latex": "$\\Sigma \\alpha_1^a \\alpha_2^a \\alpha_3^a\n = \\frac{1}{6}(s_a^3 - 3s_a s_{2a} + 2s_{3a})$,\\qquad $a > 0$.", "markdown": "$\\Sigma \\alpha_1^a \\alpha_2^a \\alpha_3^a = \\frac{1}{6}(s_a^3 - 3s_a s_{2a} + 2s_{3a})$, $a > 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:symmetric_function_identity_proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/1", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 1", "problem_latex": "$\\Sigma \\alpha_1^2 \\alpha_2 \\alpha_3$.", "markdown": "$\\Sigma \\alpha_1^2 \\alpha_2 \\alpha_3$.", "answer_latex": [ "$c_1c_3 - 4c_4$ if $n>3$, $c_1c_3$ if $n=3$." ], "answer_markdown": [ "$c_1c_3 - 4c_4$ if $n>3$, $c_1c_3$ if $n=3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c1*c3 - 4*c4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/10", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 10", "problem_latex": "$\\Sigma \\dfrac{\\beta}{\\alpha}\n = \\Sigma \\dfrac{\\beta + \\gamma + \\delta}{\\alpha}\n = \\Sigma \\dfrac{-p - \\alpha}{\\alpha}\n = -4 - p \\Sigma \\frac{1}{\\alpha}$.", "markdown": "$\\Sigma \\dfrac{\\beta}{\\alpha} = \\Sigma \\dfrac{\\beta + \\gamma + \\delta}{\\alpha} = \\Sigma \\dfrac{-p - \\alpha}{\\alpha} = -4 - p \\Sigma \\frac{1}{\\alpha}$.", "answer_latex": [ "$-4 + pr/s$." ], "answer_markdown": [ "$-4 + pr/s$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "-4 + p*r/s" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:newton_identities" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/11", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 11", "problem_latex": "$\\Sigma \\dfrac{\\beta}{\\alpha^2}$. Use\n $\\Sigma \\dfrac{1}{\\alpha}·\\Sigma \\dfrac{\\beta}{\\alpha}\n = \\Sigma \\dfrac{\\beta}{\\alpha^2}\n + 3\\Sigma \\dfrac{1}{\\alpha}\n + 2\\Sigma \\dfrac{\\gamma}{\\alpha\\beta}$.", "markdown": "$\\Sigma \\dfrac{\\beta}{\\alpha^2}$. Use $\\Sigma \\dfrac{1}{\\alpha}·\\Sigma \\dfrac{\\beta}{\\alpha} = \\Sigma \\dfrac{\\beta}{\\alpha^2} + 3\\Sigma \\dfrac{1}{\\alpha} + 2\\Sigma \\dfrac{\\gamma}{\\alpha\\beta}$.", "answer_latex": [ "$(rs - pr^2 + 2pqs)/s^2$." ], "answer_markdown": [ "$(rs - pr^2 + 2pqs)/s^2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "(r*s - p*r**2 + 2*p*q*s)/s**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:newton_identities" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/12i", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 12, "part": "i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 12i", "problem_latex": "Express $\\Sigma \\alpha_1^a \\alpha_2^b \\alpha_3^c \\alpha_4^d$ in terms of the $s_k$ when (\\emph{i})~$a>b>c>d>0$, and (\\emph{ii})~when\n$a=b=c=d$.", "markdown": "Express $\\Sigma \\alpha_1^a \\alpha_2^b \\alpha_3^c \\alpha_4^d$ in terms of the $s_k$ when (*i*) $a>b>c>d>0$, and (*ii*) when $a=b=c=d$.", "answer_latex": [ "$s_a s_b s_c s_d - \\Sigma s_a s_b s_{c+d}\n + 2\\Sigma s_a s_{b+c+d} + \\Sigma s_{a+b} s_{c+d} - 6s_{a+b+c+d}$." ], "answer_markdown": [ "$s_a s_b s_c s_d - \\Sigma s_a s_b s_{c+d} + 2\\Sigma s_a s_{b+c+d} + \\Sigma s_{a+b} s_{c+d} - 6s_{a+b+c+d}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:newton_identities" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/12ii", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 12, "part": "ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 12ii", "problem_latex": "Express $\\Sigma \\alpha_1^a \\alpha_2^b \\alpha_3^c \\alpha_4^d$ in terms of the $s_k$ when (\\emph{i})~$a>b>c>d>0$, and (\\emph{ii})~when\n$a=b=c=d$.", "markdown": "Express $\\Sigma \\alpha_1^a \\alpha_2^b \\alpha_3^c \\alpha_4^d$ in terms of the $s_k$ when (*i*) $a>b>c>d>0$, and (*ii*) when $a=b=c=d$.", "answer_latex": [ "$\\tfrac{1}{24}(s_a^4 - 6s_a^2s_{2a}\n + 8s_as_{3a} + 3s_{2a}^2 - 6s_{4a})$." ], "answer_markdown": [ "$\\tfrac{1}{24}(s_a^4 - 6s_a^2s_{2a} + 8s_as_{3a} + 3s_{2a}^2 - 6s_{4a})$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:newton_identities" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/13i", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 13, "part": "i", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 13i", "problem_latex": "By solving the first $k$ of Newton's identities~\\Eq{10} as a system of linear equations,\nfind an expression in the form of a determinant (\\emph{i})~for $s_k$ in terms of\n$c_1, \\dotsc, c_k$, and\n(\\emph{ii})~for $c_k$ in terms of $s_1, \\dotsc, s_k$.", "markdown": "By solving the first $k$ of Newton’s identities $(10)$ as a system of linear equations, find an expression in the form of a determinant (*i*) for $s_k$ in terms of $c_1, \\dotsc, c_k$, and (*ii*) for $c_k$ in terms of $s_1, \\dotsc, s_k$.", "answer_latex": [ "\\[\n s_k = - \\left|\n \\begin{array}{cccccc}\n 1 & 0 & 0 & \\ldots & 0 & c_1 \\\\\n c_1 & 1 & 0 & \\ldots & 0 & 2c_2 \\\\\n c_2 & c_1 & 1 & \\ldots & 0 & 3c_3 \\\\\n c_3 & c_2 & c_1 & \\ldots & 0 & 4c_4 \\\\\n \\Dots{6} \\\\\n c_{k-1} &c_{k-2} &c_{k-3} & \\ldots & c_1 & kc_k\n \\end{array}\\;\\right|,\\quad\n s_3 = -\n \\begin{vmatrix}\n 1 & 0 & c_1\\\\\n c_1 & 1 & 2c_2\\\\\n c_2 & c_1 & 3c_3\n \\end{vmatrix},\n \\]\n where all but the last term in the main diagonal is~$1$, and all terms above the\n diagonal are zero except those in the last column. If $k>n$, we must take\n $c_j =0 \\quad (j>n)$." ], "answer_markdown": [ "s_k = - | arraycccccc 1 & 0 & 0 & …& 0 & c_1 c_1 & 1 & 0 & …& 0 & 2c_2 c_2 & c_1 & 1 & …& 0 & 3c_3 c_3 & c_2 & c_1 & …& 0 & 4c_4 [2]6 c_k-1 &c_k-2 &c_k-3 & …& c_1 & kc_k array|, s_3 = - vmatrix 1 & 0 & c_1 c_1 & 1 & 2c_2 c_2 & c_1 & 3c_3 vmatrix, where all but the last term in the main diagonal is $1$, and all terms above the diagonal are zero except those in the last column. If $k>n$, we must take $c_j =0 \\quad (j>n)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.matrix", "core.linsys", "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/13ii", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 13, "part": "ii", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 13ii", "problem_latex": "By solving the first $k$ of Newton's identities~\\Eq{10} as a system of linear equations,\nfind an expression in the form of a determinant (\\emph{i})~for $s_k$ in terms of\n$c_1, \\dotsc, c_k$, and\n(\\emph{ii})~for $c_k$ in terms of $s_1, \\dotsc, s_k$.", "markdown": "By solving the first $k$ of Newton’s identities $(10)$ as a system of linear equations, find an expression in the form of a determinant (*i*) for $s_k$ in terms of $c_1, \\dotsc, c_k$, and (*ii*) for $c_k$ in terms of $s_1, \\dotsc, s_k$.", "answer_latex": [ "\\[\n k!\\,c_k = - \\left|\n \\begin{array}{cccccc}\n 1 & 0 & 0 & \\ldots & 0 & s_1 \\\\\n s_1 & 2 & 0 & \\ldots & 0 & s_2 \\\\\n s_2 & s_1 & 3 & \\ldots & 0 & s_3 \\\\\n \\Dots{6} \\\\\n \\ s_{k-1} & s_{k-2} & s_{k-3} & \\ldots & s_1 & s_k\n \\end{array}\\;\\right|,\\quad\n 3!\\,c_3 = -\n \\begin{vmatrix}\n 1 & 0 & s_1\\\\\n s_1 & 2 & s_2\\\\\n s_2 & s_1 & s_3\n \\end{vmatrix}.\n \\]" ], "answer_markdown": [ "k! c_k = - | arraycccccc 1 & 0 & 0 & …& 0 & s_1 s_1 & 2 & 0 & …& 0 & s_2 s_2 & s_1 & 3 & …& 0 & s_3 [2]6 s_k-1 & s_k-2 & s_k-3 & …& s_1 & s_k array|, 3! c_3 = - vmatrix 1 & 0 & s_1 s_1 & 2 & s_2 s_2 & s_1 & s_3 vmatrix." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.matrix", "core.linsys", "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/14", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 14", "problem_latex": "One set of $n$~numbers is a mere rearrangement of another set if $s_1, \\dotsc, s_n$\nhave the same values for each set.", "markdown": "One set of $n$ numbers is a mere rearrangement of another set if $s_1, \\dotsc, s_n$ have the same values for each set.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/2", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 2", "problem_latex": "$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3$.", "markdown": "$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3$.", "answer_latex": [ "$3c_1c_4 - c_2c_3 - 5c_5$." ], "answer_markdown": [ "$3c_1c_4 - c_2c_3 - 5c_5$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "3*c1*c4 - c2*c3 - 5*c5" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/3", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 3", "problem_latex": "\\makebox[0pt][l]{$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3 \\alpha_4$.}", "markdown": "[0pt][l]$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3 \\alpha_4$.", "answer_latex": [ "$c_2c_4 - 4c_1c_5 + 9c_6$." ], "answer_markdown": [ "$c_2c_4 - 4c_1c_5 + 9c_6$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c2*c4 - 4*c1*c5 + 9*c6" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/4", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 4", "problem_latex": "$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3^2$.", "markdown": "$\\Sigma \\alpha_1^2 \\alpha_2^2 \\alpha_3^2$.", "answer_latex": [ "$c_3^2 - 2c_2c_4 + 2c_1c_5 - 2c_6$." ], "answer_markdown": [ "$c_3^2 - 2c_2c_4 + 2c_1c_5 - 2c_6$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c3**2 - 2*c2*c4 + 2*c1*c5 - 2*c6" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/5", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 5", "problem_latex": "$\\alpha^2$, $\\beta^2$, $\\gamma^2$.", "markdown": "$\\alpha^2$, $\\beta^2$, $\\gamma^2$.", "answer_latex": [ "$y^3 - (p^2 - 2q)y^2 + (q^2 - 2pr)y - r^2 = 0$." ], "answer_markdown": [ "$y^3 - (p^2 - 2q)y^2 + (q^2 - 2pr)y - r^2 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "y**3 - (p**2 - 2*q)*y**2 + (q**2 - 2*p*r)*y - r**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/6", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 6", "problem_latex": "$\\alpha\\beta$, $\\alpha\\gamma$, $\\beta\\gamma$.", "markdown": "$\\alpha\\beta$, $\\alpha\\gamma$, $\\beta\\gamma$.", "answer_latex": [ "$y^3 - qy^2 + pry - r^2 = 0$." ], "answer_markdown": [ "$y^3 - qy^2 + pry - r^2 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "y**3 - q*y**2 + p*r*y - r**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/7", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 7", "problem_latex": "$\\dfrac{2}{\\alpha}$, $\\dfrac{2}{\\beta}$, $\\dfrac{2}{\\gamma}$.", "markdown": "$\\dfrac{2}{\\alpha}$, $\\dfrac{2}{\\beta}$, $\\dfrac{2}{\\gamma}$.", "answer_latex": [ "$ry^3 + 2qy^2 + 4py + 8 = 0$." ], "answer_markdown": [ "$ry^3 + 2qy^2 + 4py + 8 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "r*y**3 + 2*q*y**2 + 4*p*y + 8" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.cancel", "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/8", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 8", "problem_latex": "$\\alpha^2 + \\beta^2$, $\\alpha^2 + \\gamma^2$, $\\beta^2 + \\gamma^2$.", "markdown": "$\\alpha^2 + \\beta^2$, $\\alpha^2 + \\gamma^2$, $\\beta^2 + \\gamma^2$.", "answer_latex": [ "Eliminate $x$ by $y = s_2 - x^2$." ], "answer_markdown": [ "Eliminate $x$ by $y = s_2 - x^2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page142/9", "set": "dickson-theory-of-equations-1922/ex-page142", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "142", "location": "Exercise Page142, problem 9", "problem_latex": "$\\alpha^2 + \\alpha\\beta + \\beta^2$, etc.", "markdown": "$\\alpha^2 + \\alpha\\beta + \\beta^2$, etc.", "answer_latex": [ "Use $p^2 - q + px = y$." ], "answer_markdown": [ "Use $p^2 - q + px = y$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/1", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 1", "problem_latex": "Divide $x^3 + 3x^2 - 2x - 5$ by $x-2$.", "markdown": "Divide $x^3 + 3x^2 - 2x - 5$ by $x-2$.", "answer_latex": [ "Rem.~$11$, quot.~$x^2 + 5x + 8$." ], "answer_markdown": [ "Rem. $11$, quot. $x^2 + 5x + 8$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x**3 + 3*x**2 - 2*x - 5)/(x - 2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/2", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 2", "problem_latex": "Divide $2x^5 - x^3 + 2x - 1$ by $x+2$.", "markdown": "Divide $2x^5 - x^3 + 2x - 1$ by $x+2$.", "answer_latex": [ "$-61$, $2x^4 - 4x^3 + 7x^2 - 14x + 30$." ], "answer_markdown": [ "$-61$, $2x^4 - 4x^3 + 7x^2 - 14x + 30$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(2*x**5 - x**3 + 2*x - 1)/(x + 2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/3", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 3", "problem_latex": "Divide $x^3 + 6x^2 + 10x - 1$ by $x - 0.09$.", "markdown": "Divide $x^3 + 6x^2 + 10x - 1$ by $x - 0.09$.", "answer_latex": [ "$-0.050671$, $x^2 + 6.09x + 10.5481$." ], "answer_markdown": [ "$-0.050671$, $x^2 + 6.09x + 10.5481$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x**3 + 6*x**2 + 10*x - 1)/(x - 0.09)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/4", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 4", "problem_latex": "Find the quotient of $x^3 - 5x^2 - 2x + 24$ by $x-4$, and then divide the quotient by\n$x-3$. What are the roots of $x^3 - 5x^2 - 2x + 24 = 0$?", "markdown": "Find the quotient of $x^3 - 5x^2 - 2x + 24$ by $x-4$, and then divide the quotient by $x-3$. What are the roots of $x^3 - 5x^2 - 2x + 24 = 0$?", "answer_latex": [ "$x^2 - x - 6$, $x+2$;\\quad $4$, $3$, $-2$." ], "answer_markdown": [ "$x^2 - x - 6$, $x+2$; $4$, $3$, $-2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/5", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 5", "problem_latex": "Given that $x^4 - 2x^3 - 7x^2 + 8x + 12 = 0$ has the roots $-1$ and~$2$, find the quadratic\nequation whose roots are the remaining two roots of the given equation, and find these\nroots.", "markdown": "Given that $x^4 - 2x^3 - 7x^2 + 8x + 12 = 0$ has the roots $-1$ and $2$, find the quadratic equation whose roots are the remaining two roots of the given equation, and find these roots.", "answer_latex": [ "$x^2 - x - 6 = 0$, $3$, $-2$." ], "answer_markdown": [ "$x^2 - x - 6 = 0$, $3$, $-2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/6", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 6", "problem_latex": "If $x^4 - 2x^3 - 12x^2 + 10x + 3 = 0$ has the roots $1$ and~$-3$, find the remaining two roots.", "markdown": "If $x^4 - 2x^3 - 12x^2 + 10x + 3 = 0$ has the roots $1$ and $-3$, find the remaining two roots.", "answer_latex": [ "$2±\\sqrt{5}$." ], "answer_markdown": [ "$2±\\sqrt{5}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 2*x**3 - 12*x**2 + 10*x + 3, 0)", "answer_expr": "[2 - sqrt(5), 2 + sqrt(5)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 2*x**3 - 12*x**2 + 10*x + 3, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.pdiv", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/7", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 7", "problem_latex": "Find the quotient of $2x^4 - x^3 - 6x^2 + 4x - 8$ by $x^2 - 4$.", "markdown": "Find the quotient of $2x^4 - x^3 - 6x^2 + 4x - 8$ by $x^2 - 4$.", "answer_latex": [ "$2x^2 - x + 2$." ], "answer_markdown": [ "$2x^2 - x + 2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x**4 - x**3 - 6*x**2 + 4*x - 8)/(x**2 - 4)", "answer_expr": "2*x**2 - x + 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*x**4 - x**3 - 6*x**2 + 4*x - 8)/(x**2 - 4)" ], "shape": [ "identity: (N*x + 2*N*x**N + N - x**N)/(N + x**N)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/8", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 8", "problem_latex": "Find the quotient of $x^4 - 3x^3 + 3x^2 - 3x + 2$ by $x^2 - 3x + 2$.", "markdown": "Find the quotient of $x^4 - 3x^3 + 3x^2 - 3x + 2$ by $x^2 - 3x + 2$.", "answer_latex": [ "$x^2 + 1$." ], "answer_markdown": [ "$x^2 + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**4 - 3*x**3 + 3*x**2 - 3*x + 2)/(x**2 - 3*x + 2)", "answer_expr": "x**2 + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**4 - 3*x**3 + 3*x**2 - 3*x + 2)/(x**2 - 3*x + 2)" ], "shape": [ "identity: (N*x + 2*N*x**N + N + x**N)/(N*x + N + x**N)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page15/9", "set": "dickson-theory-of-equations-1922/ex-page15", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "15", "location": "Exercise Page15, problem 9", "problem_latex": "Solve Exercises 1, 2, 3, 6, 7 of~§14 by synthetic division.", "markdown": "Solve Exercises 1, 2, 3, 6, 7 of §14 by synthetic division.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page152/1", "set": "dickson-theory-of-equations-1922/ex-page152", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "152", "location": "Exercise Page152, problem 1", "problem_latex": "$x^2-y^2=9$, $xy = 5y$.", "markdown": "$x^2-y^2=9$, $xy = 5y$.", "answer_latex": [ "$y^2(16 - y^2)$;\\quad $y=0$, $x=±3$;\\quad $y=±4$, $x=+5$." ], "answer_markdown": [ "$y^2(16 - y^2)$; $y=0$, $x=±3$; $y=±4$, $x=+5$." ], "checks": [ { "task": "solve", "verdict": "FLAG-PARSE", "judge_why": "TypeError: unsupported operand type(s) for ** or pow(): 'Dict' and 'Integer'", "problem_expr": null, "answer_expr": "[{x: 3, y: 0}, {x: -3, y: 0}, {x: 5, y: 4}, {x: 5, y: -4}]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "solve: (Eq(a*x, 5*a), Eq(-a**2 + x**2, 9))" ], "shape": [ "solve: (Eq(a*x, N*a), Eq(-a**N + x**N, N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page152/2", "set": "dickson-theory-of-equations-1922/ex-page152", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "152", "location": "Exercise Page152, problem 2", "problem_latex": "$x^2 + y^2 = 25$, $x^2 + 3(c-1)x + c(y^2 - 25) = 0$.", "markdown": "$x^2 + y^2 = 25$, $x^2 + 3(c-1)x + c(y^2 - 25) = 0$.", "answer_latex": [ "$(c-1)^2(y^2 - 25)(y^2 - 16)$. If $c\\neq 1$, $y=±5$, $x=0$;\\quad $y=±4$, $x=+3$." ], "answer_markdown": [ "$(c-1)^2(y^2 - 25)(y^2 - 16)$. If $c\\neq 1$, $y=±5$, $x=0$; $y=±4$, $x=+3$." ], "checks": [ { "task": "solve", "verdict": "FLAG-PARSE", "judge_why": "TypeError: unsupported operand type(s) for ** or pow(): 'Dict' and 'Integer'", "problem_expr": null, "answer_expr": "[{x: 0, y: 5}, {x: 0, y: -5}, {x: 3, y: 4}, {x: 3, y: -4}]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "solve: (Eq(b**2 + x**2, 25), Eq(a*(b**2 - 25) + x**2 + x*(3*a - 3), 0))" ], "shape": [ "solve: (Eq(b**N + x**N, N), Eq(a*(N + b**N) + x*(N*a + N) + x**N, 0))" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page152/3", "set": "dickson-theory-of-equations-1922/ex-page152", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "152", "location": "Exercise Page152, problem 3", "problem_latex": "When $x^2 + ax + b = 0$ has a double root, what $3$-rowed determinant is zero?", "markdown": "When $x^2 + ax + b = 0$ has a double root, what $3$-rowed determinant is zero?", "answer_latex": [ "\n$\\begin{vmatrix}\n 1 & a & b \\\\\n 2 & a & 0 \\\\\n 0 & 2 & a\n\\end{vmatrix} = 4b-a^2$." ], "answer_markdown": [ "$\\begin{vmatrix} 1 & a & b \\\\ 2 & a & 0 \\\\ 0 & 2 & a \\end{vmatrix} = 4b-a^2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page152/4", "set": "dickson-theory-of-equations-1922/ex-page152", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "152", "location": "Exercise Page152, problem 4", "problem_latex": "Find the roots of $x^6 + 3x^4 + 32x^3 + 67x^2 + 32x + 65 = 0$ by~§79.", "markdown": "Find the roots of $x^6 + 3x^4 + 32x^3 + 67x^2 + 32x + 65 = 0$ by §79.", "answer_latex": [ "$2±3i$, $-2±i$, $±i$.\n $\\vphantom{\\begin{vmatrix}1\\\\ 1\\\\ 1\\end{vmatrix}}$" ], "answer_markdown": [ "$2±3i$, $-2±i$, $±i$. $\\vphantom{\\begin{vmatrix}1\\\\ 1\\\\ 1\\end{vmatrix}}$" ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**6 + 3*x**4 + 32*x**3 + 67*x**2 + 32*x + 65, 0) fails at {x: 3*I + 2}: leaves 96.0*I + (3.0*I + 2.0)**6 + 3.0*(3.0*I + 2.0)**4 + 32.0*(3.0*I + 2.0)**3 + 67.0*(3.0*I + 2.0)**2 + 129.0", "problem_expr": "Eq(x**6 + 3*x**4 + 32*x**3 + 67*x**2 + 32*x + 65, 0)", "answer_expr": "[2 + 3*I, 2 - 3*I, -2 + I, -2 - I, I, -I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**6 + 3*x**4 + 32*x**3 + 67*x**2 + 32*x + 65, 0)" ], "shape": [ "solve: Eq(N*x + 3*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/1", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 1", "problem_latex": "Find the equation whose roots are the abscissas of the points of intersection\nof two general conics.", "markdown": "Find the equation whose roots are the abscissas of the points of intersection of two general conics.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:resultant_elimination" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/10", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 10", "problem_latex": "Prove that the equation whose roots are the $n(n-1)$ differences $x_j-x_k$ of the\nroots of $f(x)=0$ may be obtained by eliminating $x$ between the latter and $f(x+y)=0$\nand deleting from the eliminant the factor~$y^n$ (arising from $y = x_j - x_j = 0$). The\nequation free of this factor may be obtained by eliminating~$x$ between $f(x)=0$ and\n\\index{Equation for differences of roots}%\n\\[\n\\bigl\\{f(x+y) - f(x)\\bigr\\}/y\n = f'(x) + f''(x)\\frac{y}{1·2} + \\dotsb\n + f^{(n)}(x)\\frac{y^{n-1}}{1·2\\dotsm n} = 0.\n\\]\nThis eliminant involves only even powers of~$y$, so that if we set $y^2 = z$ we obtain an\nequation in~$z$ having as its roots the squares of the differences of the roots of $f(x)=0$.\n\\index{Equation for differences of roots!squares of differences}%\n(Lagrange \\textit{Résolution des équations}, 1798,~§8.)", "markdown": "Prove that the equation whose roots are the $n(n-1)$ differences $x_j-x_k$ of the roots of $f(x)=0$ may be obtained by eliminating $x$ between the latter and $f(x+y)=0$ and deleting from the eliminant the factor $y^n$ (arising from $y = x_j - x_j = 0$). The equation free of this factor may be obtained by eliminating $x$ between $f(x)=0$ and % f(x+y) - f(x)/y = f’(x) + f”(x)y1·2 + + f^(n)(x)y^n-11·2n = 0. This eliminant involves only even powers of $y$, so that if we set $y^2 = z$ we obtain an equation in $z$ having as its roots the squares of the differences of the roots of $f(x)=0$. % (Lagrange *Résolution des équations*, 1798, §8.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "other:resultant_elimination" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/11", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 11", "problem_latex": "Compute by Ex.~10 the $z$-equation when $f(x) = x^3 + px + q$.", "markdown": "Compute by Ex. 10 the $z$-equation when $f(x) = x^3 + px + q$.", "answer_latex": [ "See Ex.~15, p.~134." ], "answer_markdown": [ "See Ex. 15, p. 134." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.derive", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/2", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 2", "problem_latex": "Find a necessary and sufficient condition that\n\\[\n f(x) \\equiv x^4 + px^3 + qx^2 + rx + s = 0\n\\]\nshall have one root the negative of another root. When this condition is satisfied,\nwhat are the quadratic factors of~$f(x)$? Apply to Ex.~4,~§74. Hint: add and subtract\n$f(x)$ and~$f(-x)$.", "markdown": "Find a necessary and sufficient condition that f(x) x^4 + px^3 + qx^2 + rx + s = 0 shall have one root the negative of another root. When this condition is satisfied, what are the quadratic factors of $f(x)$? Apply to Ex. 4, §74. Hint: add and subtract $f(x)$ and $f(-x)$.", "answer_latex": [ "$pqr - p^2s - r^2 = 0$, $x^2 + r/p$, $x^2 + px + ps/r$." ], "answer_markdown": [ "$pqr - p^2s - r^2 = 0$, $x^2 + r/p$, $x^2 + px + ps/r$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/3", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 3", "problem_latex": "Solve $f(x) \\equiv x^4 - 6x^3 + 13x^2 - 14x + 6 = 0$, given that two roots $\\alpha$ and~$\\beta$ are such\nthat $2\\alpha + \\beta = 5$. Hint: $f(x)$ and $f(5-2x)$ have a common factor.", "markdown": "Solve $f(x) \\equiv x^4 - 6x^3 + 13x^2 - 14x + 6 = 0$, given that two roots $\\alpha$ and $\\beta$ are such that $2\\alpha + \\beta = 5$. Hint: $f(x)$ and $f(5-2x)$ have a common factor.", "answer_latex": [ "$1$, $3$, $1± i$." ], "answer_markdown": [ "$1$, $3$, $1± i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**4 - 6*x**3 + 13*x**2 - 14*x + 6, 0) fails at {x: I + 1}: leaves -14.0*I + (I + 1.0)**4 - 6.0*(I + 1.0)**3 + 13.0*(I + 1.0)**2 - 8.0", "problem_expr": "Eq(x**4 - 6*x**3 + 13*x**2 - 14*x + 6, 0)", "answer_expr": "[1, 3, 1+I, 1-I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**4 - 6*x**3 + 13*x**2 - 14*x + 6, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/4", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 4", "problem_latex": "Solve $x^3 + px + q = 0$ by eliminating $x$ between it and $x^2 + vx + w = y$ by the greatest\ncommon divisor process, and choosing $v$ and~$w$ so that in the resulting cubic equation\nfor $y$ the coefficients of $y$ and~$y^2$ are zero. The next to the last step of the elimination\n%% -----File: 160.png---Folio 154-------\ngives $x$ as a rational function of~$y$. (Tschirnhausen, \\textit{Acta Erudit.}, Lipsiae,~II, 1683,\np.~204.)", "markdown": "Solve $x^3 + px + q = 0$ by eliminating $x$ between it and $x^2 + vx + w = y$ by the greatest common divisor process, and choosing $v$ and $w$ so that in the resulting cubic equation for $y$ the coefficients of $y$ and $y^2$ are zero. The next to the last step of the elimination %% -----File: 160.png---Folio 154------- gives $x$ as a rational function of $y$. (Tschirnhausen, *Acta Erudit.*, Lipsiae, II, 1683, p. 204.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**3 + p*x + q, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(a*x + b + x**3, 0)" ], "shape": [ "solve: Eq(a*x + b + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.pgcd", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/5", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 5", "problem_latex": "Find the preceding $y$-cubic as follows. Multiply $x^2 + vx + w = y$ by~$x$ and replace\n$x^3$ by~$-px-q$; then multiply the resulting quadratic equation in~$x$ by~$x$ and replace\n$x^3$ by its value. The determinant of the coefficients of $x^2$, $x$, $1$ must vanish.", "markdown": "Find the preceding $y$-cubic as follows. Multiply $x^2 + vx + w = y$ by $x$ and replace $x^3$ by $-px-q$; then multiply the resulting quadratic equation in $x$ by $x$ and replace $x^3$ by its value. The determinant of the coefficients of $x^2$, $x$, $1$ must vanish.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/6", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 6", "problem_latex": "Eliminate $y$ between $y^3 = v$, $x = ry + sy^2$, and get\n\\[\nx^3 - 3rsvx - (r^3v + s^3v^2) = 0.\n\\]\nTake $s=1$ and choose %[** PP: Typo chose]\n$r$ and~$v$ so that this equation shall be identical with $x^3 + px + q = 0$,\nand hence solve the latter. (Euler,~1764.)", "markdown": "Eliminate $y$ between $y^3 = v$, $x = ry + sy^2$, and get x^3 - 3rsvx - (r^3v + s^3v^2) = 0. Take $s=1$ and choose %[** PP: Typo chose] $r$ and $v$ so that this equation shall be identical with $x^3 + px + q = 0$, and hence solve the latter. (Euler, 1764.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/7", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 7", "problem_latex": "Eliminate $y$ between $y^3 = v$, $x = f + ey + y^2$ and get\n\\[\n\\begin{vmatrix}\n 1 & e & f-x \\\\\n e & f-x & v \\\\\n f-x & v & ev\n\\end{vmatrix}\n=0.\n\\]\nThis cubic equation in $x$ may be identified with the general cubic equation by choice\nof $e$, $f$, $v$. % [** PP: , -> .]\nHence solve the latter.", "markdown": "Eliminate $y$ between $y^3 = v$, $x = f + ey + y^2$ and get vmatrix 1 & e & f-x e & f-x & v f-x & v & ev vmatrix =0. This cubic equation in $x$ may be identified with the general cubic equation by choice of $e$, $f$, $v$. % [** PP: , -> .] Hence solve the latter.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.matrix" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/8", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 8", "problem_latex": "Determine $r$, $s$ and~$v$ so that the resultant of\n\\[\ny^3 = v,\\qquad y = \\frac{x+r}{y+s}\n\\]\nshall be identical with $x^3 + px + q = 0$. (Bézout,~1762.)", "markdown": "Determine $r$, $s$ and $v$ so that the resultant of y^3 = v, y = x+ry+s shall be identical with $x^3 + px + q = 0$. (Bézout, 1762.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pgcd", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page153/9", "set": "dickson-theory-of-equations-1922/ex-page153", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "153", "location": "Exercise Page153, problem 9", "problem_latex": "Show that the reduction of a cubic equation in~$x$ to the form $y^3 = v$ by the substitution\n\\[\nx = \\frac{r + sy}{1 + y}\n\\]\nis not essentially different from the method of Ex.~7. [Multiply the numerator and\ndenominator of~$x$ by $1 - y + y^2$.]", "markdown": "Show that the reduction of a cubic equation in $x$ to the form $y^3 = v$ by the substitution x = r + sy1 + y is not essentially different from the method of Ex. 7. [Multiply the numerator and denominator of $x$ by $1 - y + y^2$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page17/1", "set": "dickson-theory-of-equations-1922/ex-page17", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Exercise Page17, problem 1", "problem_latex": "Find a cubic equation having the roots $0$, $1$, $2$.", "markdown": "Find a cubic equation having the roots $0$, $1$, $2$.", "answer_latex": [ "$x^3 - 3x^2 + 2x = 0$." ], "answer_markdown": [ "$x^3 - 3x^2 + 2x = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page17/2", "set": "dickson-theory-of-equations-1922/ex-page17", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Exercise Page17, problem 2", "problem_latex": "Find a quartic equation having the roots $±1$, $±2$.", "markdown": "Find a quartic equation having the roots $±1$, $±2$.", "answer_latex": [ "$x^4 - 5x^2 + 4 = 0$." ], "answer_markdown": [ "$x^4 - 5x^2 + 4 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page17/3", "set": "dickson-theory-of-equations-1922/ex-page17", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Exercise Page17, problem 3", "problem_latex": "Find a quartic equation having the two double roots $3$ and~$-3$.", "markdown": "Find a quartic equation having the two double roots $3$ and $-3$.", "answer_latex": [ "$x^4 - 18x^2 + 81 = 0$." ], "answer_markdown": [ "$x^4 - 18x^2 + 81 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page17/4", "set": "dickson-theory-of-equations-1922/ex-page17", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Exercise Page17, problem 4", "problem_latex": "Find a quartic equation having the root~$2$ and the triple root~$1$.", "markdown": "Find a quartic equation having the root $2$ and the triple root $1$.", "answer_latex": [ "$x^4 - 5x^3 + 9x^2 - 7x + 2 = 0$." ], "answer_markdown": [ "$x^4 - 5x^3 + 9x^2 - 7x + 2 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page17/5", "set": "dickson-theory-of-equations-1922/ex-page17", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Exercise Page17, problem 5", "problem_latex": "What is the condition that $ax^2+bx+c=0$ shall have a double root?", "markdown": "What is the condition that $ax^2+bx+c=0$ shall have a double root?", "answer_latex": [ "$b^2 = 4ac$." ], "answer_markdown": [ "$b^2 = 4ac$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page17/6", "set": "dickson-theory-of-equations-1922/ex-page17", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Exercise Page17, problem 6", "problem_latex": "If $a_0 x^n + \\dotsb + a_n = 0$ has more than $n$~distinct roots, each coefficient is zero.", "markdown": "If $a_0 x^n + \\dotsb + a_n = 0$ has more than $n$ distinct roots, each coefficient is zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page17/7", "set": "dickson-theory-of-equations-1922/ex-page17", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "17", "location": "Exercise Page17, problem 7", "problem_latex": "Why is there a single answer to each of Exercises 1--4, if the coefficient of the\nhighest power of the unknown be taken equal to unity? State and answer the corresponding\ngeneral question.", "markdown": "Why is there a single answer to each of Exercises 1--4, if the coefficient of the highest power of the unknown be taken equal to unity? State and answer the corresponding general question.", "answer_latex": [ "By theorem in~§18." ], "answer_markdown": [ "By theorem in §18." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/1", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 1", "problem_latex": "Find a cubic equation having the roots $1$, $2$, $3$.", "markdown": "Find a cubic equation having the roots $1$, $2$, $3$.", "answer_latex": [ "$x^3 - 6x^2 + 11x - 6 = 0$." ], "answer_markdown": [ "$x^3 - 6x^2 + 11x - 6 = 0$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x-1)*(x-2)*(x-3)", "answer_expr": "x**3 - 6*x**2 + 11*x - 6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x - 3)*(x - 2)*(x - 1)" ], "shape": [ "identity: (N + x)**2*(x - 1)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/10", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 10", "problem_latex": "Solve $x^4 - 2x^3 - 21x^2 + 22x + 40 = 0$, whose roots are in arithmetical progression.\n[Denote them by $c-3b$, $c-b$, $c+b$, $c+3b$, with the common difference $2b$]. % [** PP: Added period]", "markdown": "Solve $x^4 - 2x^3 - 21x^2 + 22x + 40 = 0$, whose roots are in arithmetical progression. [Denote them by $c-3b$, $c-b$, $c+b$, $c+3b$, with the common difference $2b$]. % [** PP: Added period]", "answer_latex": [ "$5$, $2$, $-1$, $-4$." ], "answer_markdown": [ "$5$, $2$, $-1$, $-4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 2*x**3 - 21*x**2 + 22*x + 40, 0)", "answer_expr": "[5, 2, -1, -4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 2*x**3 - 21*x**2 + 22*x + 40, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page27/1" ], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/11", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 11", "problem_latex": "Find a quadratic equation whose roots are the squares of the roots of\n$x^2-px+q = 0$.", "markdown": "Find a quadratic equation whose roots are the squares of the roots of $x^2-px+q = 0$.", "answer_latex": [ "$y^2 - (p^2 - 2q)y + q^2 = 0$." ], "answer_markdown": [ "$y^2 - (p^2 - 2q)y + q^2 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/12", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 12", "problem_latex": "Find a quadratic equation whose roots are the cubes of the roots of $x^2 - px + q = 0$.\nHint: $\\alpha^3 + \\beta^3 = (\\alpha+\\beta)^3 - 3\\alpha\\beta(\\alpha+\\beta)$.", "markdown": "Find a quadratic equation whose roots are the cubes of the roots of $x^2 - px + q = 0$. Hint: $\\alpha^3 + \\beta^3 = (\\alpha+\\beta)^3 - 3\\alpha\\beta(\\alpha+\\beta)$.", "answer_latex": [ "$y^2 - (p^3 - 3pq)y + q^3 = 0$." ], "answer_markdown": [ "$y^2 - (p^3 - 3pq)y + q^3 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/13a", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 13, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 13a", "problem_latex": "If $\\alpha$ and~$\\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i)~$\\alpha^2 / \\beta$;\n and~$\\beta^2 / \\alpha$; (ii)~$\\alpha^3\\beta$ and~$\\alpha\\beta^3$; (iii)~$\\alpha+1 / \\beta$ and~$\\beta + 1 / \\alpha$.", "markdown": "If $\\alpha$ and $\\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i) $\\alpha^2 / \\beta$; and $\\beta^2 / \\alpha$; (ii) $\\alpha^3\\beta$ and $\\alpha\\beta^3$; (iii) $\\alpha+1 / \\beta$ and $\\beta + 1 / \\alpha$.", "answer_latex": [ "$y^2 - y(p^3 - 3pq)/q + q = 0$." ], "answer_markdown": [ "$y^2 - y(p^3 - 3pq)/q + q = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/13b", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 13, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 13b", "problem_latex": "If $\\alpha$ and~$\\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i)~$\\alpha^2 / \\beta$;\n and~$\\beta^2 / \\alpha$; (ii)~$\\alpha^3\\beta$ and~$\\alpha\\beta^3$; (iii)~$\\alpha+1 / \\beta$ and~$\\beta + 1 / \\alpha$.", "markdown": "If $\\alpha$ and $\\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i) $\\alpha^2 / \\beta$; and $\\beta^2 / \\alpha$; (ii) $\\alpha^3\\beta$ and $\\alpha\\beta^3$; (iii) $\\alpha+1 / \\beta$ and $\\beta + 1 / \\alpha$.", "answer_latex": [ "$y^2 - q(p^2 - 2q)y + q^4 = 0$." ], "answer_markdown": [ "$y^2 - q(p^2 - 2q)y + q^4 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/13c", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 13, "part": "c", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 13c", "problem_latex": "If $\\alpha$ and~$\\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i)~$\\alpha^2 / \\beta$;\n and~$\\beta^2 / \\alpha$; (ii)~$\\alpha^3\\beta$ and~$\\alpha\\beta^3$; (iii)~$\\alpha+1 / \\beta$ and~$\\beta + 1 / \\alpha$.", "markdown": "If $\\alpha$ and $\\beta$ are the roots of $x^2 - px + q = 0$, find an equation whose roots are (i) $\\alpha^2 / \\beta$; and $\\beta^2 / \\alpha$; (ii) $\\alpha^3\\beta$ and $\\alpha\\beta^3$; (iii) $\\alpha+1 / \\beta$ and $\\beta + 1 / \\alpha$.", "answer_latex": [ "$y^2 - (p + p/q)y + 2 + q + 1/q = 0$." ], "answer_markdown": [ "$y^2 - (p + p/q)y + 2 + q + 1/q = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/14", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 14", "problem_latex": "Find a necessary and sufficient condition that the roots, taken in some order,\nof $x^3 + px^2 + qx + r = 0$ shall be in geometrical progression.", "markdown": "Find a necessary and sufficient condition that the roots, taken in some order, of $x^3 + px^2 + qx + r = 0$ shall be in geometrical progression.", "answer_latex": [ "$p^3r = q^3$." ], "answer_markdown": [ "$p^3r = q^3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/15", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 15, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 15", "problem_latex": "Solve $x^3 - 28x + 48 = 0$, given that two roots differ by~$2$.", "markdown": "Solve $x^3 - 28x + 48 = 0$, given that two roots differ by $2$.", "answer_latex": [ "$2$, $4$, $-6$." ], "answer_markdown": [ "$2$, $4$, $-6$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 28*x + 48, 0)", "answer_expr": "[2, 4, -6]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 28*x + 48, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/2", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 2", "problem_latex": "Find a quartic equation having the double roots $2$ and~$-2$.", "markdown": "Find a quartic equation having the double roots $2$ and $-2$.", "answer_latex": [ "$x^4 - 8x^2 + 16 = 0$." ], "answer_markdown": [ "$x^4 - 8x^2 + 16 = 0$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x-2)**2*(x+2)**2", "answer_expr": "x**4 - 8*x**2 + 16" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x - 2)**2*(x + 2)**2" ], "shape": [ "identity: (N + x)**(2*N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/3", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 3", "problem_latex": "Solve $x^4 - 6x^3 + 13x^2 - 12x + 4 = 0$, which has two double roots.", "markdown": "Solve $x^4 - 6x^3 + 13x^2 - 12x + 4 = 0$, which has two double roots.", "answer_latex": [ "$1$, $2$." ], "answer_markdown": [ "$1$, $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 6*x**3 + 13*x**2 - 12*x + 4, 0)", "answer_expr": "[1, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 6*x**3 + 13*x**2 - 12*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/4", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 4", "problem_latex": "Prove that one root of $x^3 + px^2 + qx + r = 0$ is the negative of another root if and\nonly if $r = pq$.", "markdown": "Prove that one root of $x^3 + px^2 + qx + r = 0$ is the negative of another root if and only if $r = pq$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/5", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 5", "problem_latex": "Solve $4x^3 - 16x^2 - 9x + 36 = 0$, given that one root is the negative of another.", "markdown": "Solve $4x^3 - 16x^2 - 9x + 36 = 0$, given that one root is the negative of another.", "answer_latex": [ "$4$, $\\tfrac{3}{2}$, $-\\tfrac{3}{2}$." ], "answer_markdown": [ "$4$, $\\tfrac{3}{2}$, $-\\tfrac{3}{2}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4*x**3 - 16*x**2 - 9*x + 36, 0)", "answer_expr": "[4, Rational(3, 2), Rational(-3, 2)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(4*x**3 - 16*x**2 - 9*x + 36, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/6", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 6", "problem_latex": "Solve $x^3 - 9x^2 + 23x - 15 = 0$, given that one root is the triple of another.", "markdown": "Solve $x^3 - 9x^2 + 23x - 15 = 0$, given that one root is the triple of another.", "answer_latex": [ "$1$, $3$, $5$." ], "answer_markdown": [ "$1$, $3$, $5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 9*x**2 + 23*x - 15, 0)", "answer_expr": "[1, 3, 5]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 9*x**2 + 23*x - 15, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/7", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 7", "problem_latex": "Solve $x^4 - 6x^3 + 12x^2 - 10x + 3 = 0$, which has a triple root.", "markdown": "Solve $x^4 - 6x^3 + 12x^2 - 10x + 3 = 0$, which has a triple root.", "answer_latex": [ "$1$, $1$, $1$, $3$." ], "answer_markdown": [ "$1$, $1$, $1$, $3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 6*x**3 + 12*x**2 - 10*x + 3, 0)", "answer_expr": "[1, 1, 1, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 6*x**3 + 12*x**2 - 10*x + 3, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/8", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 8", "problem_latex": "Solve $x^3 - 14x^2 - 84x + 216 = 0$, whose roots are in geometrical progression, i.e.,\nwith a common ratio $r$ [say $m/r$, $m$, $mr$].", "markdown": "Solve $x^3 - 14x^2 - 84x + 216 = 0$, whose roots are in geometrical progression, i.e., with a common ratio $r$ [say $m/r$, $m$, $mr$].", "answer_latex": [ "$2$, $-6$, $18$." ], "answer_markdown": [ "$2$, $-6$, $18$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 14*x**2 - 84*x + 216, 0)", "answer_expr": "[2, -6, 18]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 14*x**2 - 84*x + 216, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page19/9", "set": "dickson-theory-of-equations-1922/ex-page19", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "19", "location": "Exercise Page19, problem 9", "problem_latex": "Solve $x^3 - 3x^2 - 13x + 15 = 0$, whose roots are in arithmetical progression, i.e.,\nwith a common difference $d$ [say $m-d$, $m$, $m+d$].", "markdown": "Solve $x^3 - 3x^2 - 13x + 15 = 0$, whose roots are in arithmetical progression, i.e., with a common difference $d$ [say $m-d$, $m$, $m+d$].", "answer_latex": [ "$-3$, $1$, $5$." ], "answer_markdown": [ "$-3$, $1$, $5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 3*x**2 - 13*x + 15, 0)", "answer_expr": "[-3, 1, 5]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 3*x**2 - 13*x + 15, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/1", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 1", "problem_latex": "$\\sqrt{-9}$.", "markdown": "$\\sqrt{-9}$.", "answer_latex": [ "$3i$." ], "answer_markdown": [ "$3i$." ], "checks": [ { "task": "identity", "verdict": "FLAG-PARSE", "judge_why": "undeclared symbols ['I']", "problem_expr": "sqrt(-9)", "answer_expr": "3*I" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "identity: 3*I" ], "shape": [ "identity: I*N" ], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/10", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 10", "problem_latex": "Prove that the conjugate of the sum of two complex numbers is equal to the\nsum of their conjugates. Does the result hold true if each word sum is replaced by the\nword difference?", "markdown": "Prove that the conjugate of the sum of two complex numbers is equal to the sum of their conjugates. Does the result hold true if each word sum is replaced by the word difference?", "answer_latex": [ "Yes." ], "answer_markdown": [ "Yes." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/11", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 11", "problem_latex": "Prove that the conjugate of the product (or quotient) of two complex numbers\nis equal to the product (or quotient) of their conjugates.", "markdown": "Prove that the conjugate of the product (or quotient) of two complex numbers is equal to the product (or quotient) of their conjugates.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/12", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 12", "problem_latex": "Prove that, if the product of two complex numbers is zero, at least one of them\nis zero.", "markdown": "Prove that, if the product of two complex numbers is zero, at least one of them is zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/13", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 13, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 13", "problem_latex": "Find two pairs of real numbers $x$, $y$ for which\n\\[\n(x+yi)^2 = -7+24i.\n\\]", "markdown": "Find two pairs of real numbers $x$, $y$ for which (x+yi)^2 = -7+24i.", "answer_latex": [ "$3$, $4$ and $-3$, $-4$." ], "answer_markdown": [ "$3$, $4$ and $-3$, $-4$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "Eq((x + y*I)**2, -7 + 24*I)", "answer_expr": "[(3, 4), (-3, -4)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.nonpoly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/14", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 14", "problem_latex": "{$-11+60i$.}", "markdown": "$-11+60i$.", "answer_latex": [ "$±(5 + 6i)$." ], "answer_markdown": [ "$±(5 + 6i)$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(z**2, 60*I - 11) fails at {z: 6*I + 5}: ('50.578512396694214876', '0.0')", "problem_expr": "Eq(z**2, -11 + 60*I)", "answer_expr": "[5 + 6*I, -5 - 6*I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**2, 60*a - 11)" ], "shape": [ "solve: Eq(x**N, N*a + N)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/15", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 15, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 15", "problem_latex": "{$5-12i$.}", "markdown": "$5-12i$.", "answer_latex": [ "$±(3 - 2i)$." ], "answer_markdown": [ "$±(3 - 2i)$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(z**2, 5 - 12*I) fails at {z: 3 - 2*I}: ('6.25', '0.0')", "problem_expr": "Eq(z**2, 5 - 12*I)", "answer_expr": "[3 - 2*I, -3 + 2*I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**2, -12*a + 5)" ], "shape": [ "solve: Eq(x**N, N*a + N)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/16", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 16, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 16", "problem_latex": "{$4cd+(2c^2-2d^2)i$.}", "markdown": "$4cd+(2c^2-2d^2)i$.", "answer_latex": [ "$±\\bigl[c + d + (c - d)i\\bigr]$." ], "answer_markdown": [ "$±\\bigl[c + d + (c - d)i\\bigr]$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(z**2, I*(2*c**2 - 2*d**2) + 4*c*d) fails at {z: I*(c - d) + c + d}: ('0.054713197415243894606', '0.0')", "problem_expr": "Eq(z**2, 4*c*d + (2*c**2 - 2*d**2)*I)", "answer_expr": "[c + d + (c - d)*I, -c - d - (c - d)*I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**2, a*(2*b**2 - 2*c**2) + 4*b*c)" ], "shape": [ "solve: Eq(x**N, N*b*c + a*(N*b**N + N*c**N))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.complex", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/2", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 2", "problem_latex": "$\\sqrt{4}$.", "markdown": "$\\sqrt{4}$.", "answer_latex": [ "$2$." ], "answer_markdown": [ "$2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "sqrt(4)", "answer_expr": "2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/10", "wentworth-first-steps-in-algebra-1894/ex-1/13" ], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/3", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 3", "problem_latex": "$(\\sqrt{25} + \\sqrt{-25})\\sqrt{-16}$.", "markdown": "$(\\sqrt{25} + \\sqrt{-25})\\sqrt{-16}$.", "answer_latex": [ "$-20 + 20i$." ], "answer_markdown": [ "$-20 + 20i$." ], "checks": [ { "task": "identity", "verdict": "FLAG-PARSE", "judge_why": "undeclared symbols ['I']", "problem_expr": "(sqrt(25) + sqrt(-25))*sqrt(-16)", "answer_expr": "-20 + 20*I" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "identity: 4*I*(5 + 5*I)" ], "shape": [ "identity: I*N*(N + I*N)" ], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/4", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 4", "problem_latex": "$-\\frac{2}{3}$.", "markdown": "$-\\frac{2}{3}$.", "answer_latex": [ "$-\\frac{2}{3}$." ], "answer_markdown": [ "$-\\frac{2}{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "-Rational(2, 3)", "answer_expr": "-Rational(2, 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -2/3" ], "shape": [ "identity: N" ], "same_problem_in": [], "needs": [ "core.complex", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/5", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 5", "problem_latex": "$8 + 2\\sqrt{3}\\vphantom{\\dfrac{1}{1}}$.", "markdown": "$8 + 2\\sqrt{3}\\vphantom{\\dfrac{1}{1}}$.", "answer_latex": [ "$(8 + 2\\sqrt{3})$." ], "answer_markdown": [ "$(8 + 2\\sqrt{3})$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "8 + 2*sqrt(3)", "answer_expr": "8 + 2*sqrt(3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*sqrt(3) + 8" ], "shape": [ "identity: N*N**N + N" ], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/6", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 6", "problem_latex": "$\\dfrac{3 + \\sqrt{-5}}{2 + \\sqrt{-1}}$.", "markdown": "$\\dfrac{3 + \\sqrt{-5}}{2 + \\sqrt{-1}}$.", "answer_latex": [ "$\\frac{1}{5}(6 + \\sqrt{5}) + \\frac{1}{5}(2\\sqrt{5} - 3)i$." ], "answer_markdown": [ "$\\frac{1}{5}(6 + \\sqrt{5}) + \\frac{1}{5}(2\\sqrt{5} - 3)i$." ], "checks": [ { "task": "identity", "verdict": "FLAG-PARSE", "judge_why": "undeclared symbols ['I']", "problem_expr": "(3 + sqrt(-5))/(2 + sqrt(-1))", "answer_expr": "(6 + sqrt(5))/5 + (2*sqrt(5) - 3)*I/5" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "identity: (2 - I)*(3 + sqrt(5)*I)/5" ], "shape": [ "identity: N*(N - I)*(N + I*N**N)" ], "same_problem_in": [], "needs": [ "core.complex", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/7", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 7", "problem_latex": "$\\dfrac{3 + 5i}{2 - 3i}$.", "markdown": "$\\dfrac{3 + 5i}{2 - 3i}$.", "answer_latex": [ "$\\dfrac{-9}{13} + \\dfrac{19}{13} i$." ], "answer_markdown": [ "$\\dfrac{-9}{13} + \\dfrac{19}{13} i$." ], "checks": [ { "task": "identity", "verdict": "FLAG-PARSE", "judge_why": "undeclared symbols ['I']", "problem_expr": "(3 + 5*I)/(2 - 3*I)", "answer_expr": "-Rational(9, 13) + Rational(19, 13)*I" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "identity: (5*a + 3)/(-3*a + 2)" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "core.complex", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/8", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 8", "problem_latex": "$\\dfrac{a + bi}{a - bi}$.", "markdown": "$\\dfrac{a + bi}{a - bi}$.", "answer_latex": [ "$\\dfrac{a^2 - b^2}{a^2 + b^2} + \\dfrac{2ab}{a^2 + b^2}i$." ], "answer_markdown": [ "$\\dfrac{a^2 - b^2}{a^2 + b^2} + \\dfrac{2ab}{a^2 + b^2}i$." ], "checks": [ { "task": "identity", "verdict": "FLAG-PARSE", "judge_why": "undeclared symbols ['I']", "problem_expr": "(a + b*I)/(a - b*I)", "answer_expr": "(a**2 - b**2)/(a**2 + b**2) + 2*a*b*I/(a**2 + b**2)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "identity: (a*c + b)/(-a*c + b)" ], "shape": [ "identity: (a*c + b)/(-a*c + b)" ], "same_problem_in": [], "needs": [ "cas.expand", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page2/9", "set": "dickson-theory-of-equations-1922/ex-page2", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "2", "location": "Exercise Page2, problem 9", "problem_latex": "Prove that the sum of two conjugate complex numbers is real and that their\ndifference is a pure imaginary.", "markdown": "Prove that the sum of two conjugate complex numbers is real and that their difference is a pure imaginary.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/1", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 1", "problem_latex": "Solve $x^3 - 3x^2 - 6x - 20 = 0$, one root being $-1 + \\sqrt{-3}$.", "markdown": "Solve $x^3 - 3x^2 - 6x - 20 = 0$, one root being $-1 + \\sqrt{-3}$.", "answer_latex": [ "$5$, $-1±\\sqrt{-3}$." ], "answer_markdown": [ "$5$, $-1±\\sqrt{-3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 3*x**2 - 6*x - 20, 0)", "answer_expr": "[5, -1 + sqrt(-3), -1 - sqrt(-3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 3*x**2 - 6*x - 20, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.pdiv", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/10", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 10", "problem_latex": "Given that $x^4 - 2x^3 - 5x^2 - 6x + 2 = 0$ has the root $2 - \\sqrt{3}$, find another root and\nby means of the sum and the product of the four roots deduce, without division, the\nquadratic equation satisfied by the remaining two roots.", "markdown": "Given that $x^4 - 2x^3 - 5x^2 - 6x + 2 = 0$ has the root $2 - \\sqrt{3}$, find another root and by means of the sum and the product of the four roots deduce, without division, the quadratic equation satisfied by the remaining two roots.", "answer_latex": [ "$2 + \\sqrt{3}$, $x^2 + 2x + 2 = 0$." ], "answer_markdown": [ "$2 + \\sqrt{3}$, $x^2 + 2x + 2 = 0$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 2*x**3 - 5*x**2 - 6*x + 2, 0)", "answer_expr": "[2 + sqrt(3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 2*x**3 - 5*x**2 - 6*x + 2, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/11", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 11", "problem_latex": "Granted that a certain cubic equation has the root~$2$ and no real root different\nfrom~$2$, does it have two imaginary roots?", "markdown": "Granted that a certain cubic equation has the root $2$ and no real root different from $2$, does it have two imaginary roots?", "answer_latex": [ "Not necessarily." ], "answer_markdown": [ "Not necessarily." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/12", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 12", "problem_latex": "Granted that a certain quartic equation has the roots $2 ± 3i$, and no imaginary\nroots different from them, does it have two real roots?", "markdown": "Granted that a certain quartic equation has the roots $2 ± 3i$, and no imaginary roots different from them, does it have two real roots?", "answer_latex": [ "Not necessarily." ], "answer_markdown": [ "Not necessarily." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/13", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 13, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 13", "problem_latex": "By means of the proof of Ex.~5, may we conclude as at the end of~§21 that\nevery integral rational function with rational coefficients can be expressed as a product\nof linear and quadratic factors with rational coefficients?", "markdown": "By means of the proof of Ex. 5, may we conclude as at the end of §21 that every integral rational function with rational coefficients can be expressed as a product of linear and quadratic factors with rational coefficients?", "answer_latex": [ "No." ], "answer_markdown": [ "No." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/2", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 2", "problem_latex": "Solve $x^4 - 4x^3 + 5x^2 - 2x - 2 = 0$, one root being $1-i$.", "markdown": "Solve $x^4 - 4x^3 + 5x^2 - 2x - 2 = 0$, one root being $1-i$.", "answer_latex": [ "$1±i$, $1±\\sqrt{2}$." ], "answer_markdown": [ "$1±i$, $1±\\sqrt{2}$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**4 - 4*x**3 + 5*x**2 - 2*x - 2, 0) fails at {x: I + 1}: leaves -2.0*I + (I + 1.0)**4 - 4.0*(I + 1.0)**3 + 5.0*(I + 1.0)**2 - 4.0", "problem_expr": "Eq(x**4 - 4*x**3 + 5*x**2 - 2*x - 2, 0)", "answer_expr": "[1 + I, 1 - I, 1 + sqrt(2), 1 - sqrt(2)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**4 - 4*x**3 + 5*x**2 - 2*x - 2, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.pdiv", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/3", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 3", "problem_latex": "Find a cubic equation with real coefficients two of whose roots are $1$ and $3+2i$.", "markdown": "Find a cubic equation with real coefficients two of whose roots are $1$ and $3+2i$.", "answer_latex": [ "$x^3 - 7x^2 + 19x - 13 = 0$." ], "answer_markdown": [ "$x^3 - 7x^2 + 19x - 13 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x**3 - 7*x**2 + 19*x - 13" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/4", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 4", "problem_latex": "If a real cubic equation $x^3 - 6x^2 + \\dotsb = 0$ has the root $1 +\\sqrt{-5}$, what are the\nremaining roots? Find the complete equation.", "markdown": "If a real cubic equation $x^3 - 6x^2 + \\dotsb = 0$ has the root $1 +\\sqrt{-5}$, what are the remaining roots? Find the complete equation.", "answer_latex": [ "$4$, $1-\\sqrt{-5}$, $x^3 - 6x^2 + 14x - 24 = 0$." ], "answer_markdown": [ "$4$, $1-\\sqrt{-5}$, $x^3 - 6x^2 + 14x - 24 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/5", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 5", "problem_latex": "If an equation with \\emph{rational} coefficients has a root $a + \\sqrt{b}$, where $a$ and $b$ are\nrational, but $\\sqrt{b}$ is irrational, prove that it has the root $a - \\sqrt{b}$. [Use the method of~§21.]", "markdown": "If an equation with *rational* coefficients has a root $a + \\sqrt{b}$, where $a$ and $b$ are rational, but $\\sqrt{b}$ is irrational, prove that it has the root $a - \\sqrt{b}$. [Use the method of §21.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/6", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 6", "problem_latex": "Solve $x^4 - 4x^3 + 4x - 1 = 0$, one root being $2 + \\sqrt{3}$.", "markdown": "Solve $x^4 - 4x^3 + 4x - 1 = 0$, one root being $2 + \\sqrt{3}$.", "answer_latex": [ "$± 1$, $2±\\sqrt{3}$." ], "answer_markdown": [ "$± 1$, $2±\\sqrt{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 4*x**3 + 4*x - 1, 0)", "answer_expr": "[1, -1, 2 + sqrt(3), 2 - sqrt(3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 4*x**3 + 4*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + x**N - 1, 0)" ], "same_problem_in": [], "needs": [ "cas.pdiv", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/7", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 7", "problem_latex": "Solve $x^3 - (4 + \\sqrt{3})x^2 + (5 + 4\\sqrt{3})x - 5\\sqrt{3} = 0$, having the root $\\sqrt{3}$.", "markdown": "Solve $x^3 - (4 + \\sqrt{3})x^2 + (5 + 4\\sqrt{3})x - 5\\sqrt{3} = 0$, having the root $\\sqrt{3}$.", "answer_latex": [ "$\\sqrt{3}$, $2±i$." ], "answer_markdown": [ "$\\sqrt{3}$, $2±i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**3 - x**2*(sqrt(3) + 4) + x*(5 + 4*sqrt(3)) - 5*sqrt(3), 0) fails at {x: I + 2}: leaves 11.9282032303*I + (I + 2.0)**3 - 5.73205080757*(I + 2.0)**2 + 15.1961524227", "problem_expr": "Eq(x**3 - (4 + sqrt(3))*x**2 + (5 + 4*sqrt(3))*x - 5*sqrt(3), 0)", "answer_expr": "[sqrt(3), 2 + I, 2 - I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**3 - x**2*(sqrt(3) + 4) + x*(5 + 4*sqrt(3)) - 5*sqrt(3), 0)" ], "shape": [ "solve: Eq(N*N**N + x*(N*N**N + N) - x**N*(N + N**N) + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page20/8" ], "needs": [ "cas.pdiv", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/8", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 8", "problem_latex": "Solve the equation in Ex.~7, given that it has the root $2+i$.", "markdown": "Solve the equation in Ex. 7, given that it has the root $2+i$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**3 - (4 + sqrt(3))*x**2 + (5 + 4*sqrt(3))*x - 5*sqrt(3), 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**3 - x**2*(sqrt(3) + 4) + x*(5 + 4*sqrt(3)) - 5*sqrt(3), 0)" ], "shape": [ "solve: Eq(N*N**N + x*(N*N**N + N) - x**N*(N + N**N) + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page20/7" ], "needs": [ "cas.pdiv", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page20/9", "set": "dickson-theory-of-equations-1922/ex-page20", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "20", "location": "Exercise Page20, problem 9", "problem_latex": "Find a cubic equation with rational coefficients having the roots $\\frac{1}{2}, \\frac{1}{2} + \\sqrt{2}$.", "markdown": "Find a cubic equation with rational coefficients having the roots $\\frac{1}{2}, \\frac{1}{2} + \\sqrt{2}$.", "answer_latex": [ "$x^3 - \\tfrac{3}{2}x^2 - \\tfrac{5}{4}x + \\tfrac{7}{8} = 0$." ], "answer_markdown": [ "$x^3 - \\tfrac{3}{2}x^2 - \\tfrac{5}{4}x + \\tfrac{7}{8} = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x**3 - Rational(3, 2)*x**2 - Rational(5, 4)*x + Rational(7, 8)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page23/1", "set": "dickson-theory-of-equations-1922/ex-page23", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Exercise Page23, problem 1", "problem_latex": "$4x^5 - 8x^4 + 22x^3 + 98x^2 - 73x + 5 = 0$.", "markdown": "$4x^5 - 8x^4 + 22x^3 + 98x^2 - 73x + 5 = 0$.", "answer_latex": [ "$19\\tfrac{1}{4}$, $3$." ], "answer_markdown": [ "$19\\tfrac{1}{4}$, $3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:root_bound_method" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page23/2", "set": "dickson-theory-of-equations-1922/ex-page23", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Exercise Page23, problem 2", "problem_latex": "$x^4 - 5x^3 + 7x^2 - 8x + 1 = 0$.", "markdown": "$x^4 - 5x^3 + 7x^2 - 8x + 1 = 0$.", "answer_latex": [ "$6$." ], "answer_markdown": [ "$6$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:root_bound_method" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page23/3", "set": "dickson-theory-of-equations-1922/ex-page23", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Exercise Page23, problem 3", "problem_latex": "$x^7 + 3x^6 - 4x^5 + 5x^4 - 6x^3 - 7x^2 - 8 = 0$.", "markdown": "$x^7 + 3x^6 - 4x^5 + 5x^4 - 6x^3 - 7x^2 - 8 = 0$.", "answer_latex": [ "$2$." ], "answer_markdown": [ "$2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:root_bound_method" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page23/4", "set": "dickson-theory-of-equations-1922/ex-page23", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Exercise Page23, problem 4", "problem_latex": "$x^7 + 2x^5 + 4x^4 - 8x^2 - 32 = 0$.", "markdown": "$x^7 + 2x^5 + 4x^4 - 8x^2 - 32 = 0$.", "answer_latex": [ "$3$." ], "answer_markdown": [ "$3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:root_bound_method" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page23/5", "set": "dickson-theory-of-equations-1922/ex-page23", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Exercise Page23, problem 5", "problem_latex": "A lower limit to the negative roots of $f(x) = 0$ may be found by applying our\ntheorems to $f(-x) = 0$, i.e., to the equation derived from $f(x) = 0$ by replacing~$x$ by~$-x$.\nFind a lower limit to the negative roots in Exs.\\ 2, 3,~4.", "markdown": "A lower limit to the negative roots of $f(x) = 0$ may be found by applying our theorems to $f(-x) = 0$, i.e., to the equation derived from $f(x) = 0$ by replacing $x$ by $-x$. Find a lower limit to the negative roots in Exs. 2, 3, 4.", "answer_latex": [ "$0, -7, -\\tfrac{7}{3}$." ], "answer_markdown": [ "$0, -7, -\\tfrac{7}{3}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:root_bound_method" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page23/6", "set": "dickson-theory-of-equations-1922/ex-page23", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Exercise Page23, problem 6", "problem_latex": "Prove that every real root of a real equation $f(x) = 0$ is less than $1 + g / a_0$ if $a_0 > 0$,\nwhere $g$ denotes the greatest of the numerical values of $a_1, \\dotsc, a_n$. Hint: if $x>0$,\n\\[\na_0 x^n + a_1 x^{n-1} + \\dotsb \\geqq a_0 x^n - g(x^{n-1} + \\dotsb + x + 1).\n\\]\nProceed as in~§22 with $k = 1$.", "markdown": "Prove that every real root of a real equation $f(x) = 0$ is less than $1 + g / a_0$ if $a_0 > 0$, where $g$ denotes the greatest of the numerical values of $a_1, \\dotsc, a_n$. Hint: if $x>0$, a_0 x^n + a_1 x^n-1 + a_0 x^n - g(x^n-1 + + x + 1). Proceed as in §22 with $k = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page23/7", "set": "dickson-theory-of-equations-1922/ex-page23", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "23", "location": "Exercise Page23, problem 7", "problem_latex": "Prove that $1 + g \\div |a_0|$ is an upper limit for the moduli of all complex roots of any\nequation $f(x)=0$ with complex coefficients, where $g$ is the greatest of the values $|a_1|,\n\\dotsc, |a_n|$, and $|a|$ denotes the modulus of~$a$. Hint: use Ex.~5 of~§8.", "markdown": "Prove that $1 + g \\div |a_0|$ is an upper limit for the moduli of all complex roots of any equation $f(x)=0$ with complex coefficients, where $g$ is the greatest of the values $|a_1|, \\dotsc, |a_n|$, and $|a|$ denotes the modulus of $a$. Hint: use Ex. 5 of §8.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page25/1", "set": "dickson-theory-of-equations-1922/ex-page25", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "25", "location": "Exercise Page25, problem 1", "problem_latex": "$x^3 + 8x^2 + 13x + 6 = 0$.", "markdown": "$x^3 + 8x^2 + 13x + 6 = 0$.", "answer_latex": [ "$-1$, $-1$, $-6$." ], "answer_markdown": [ "$-1$, $-1$, $-6$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 8*x**2 + 13*x + 6, 0)", "answer_expr": "[-1, -1, -6]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 8*x**2 + 13*x + 6, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page25/2", "set": "dickson-theory-of-equations-1922/ex-page25", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "25", "location": "Exercise Page25, problem 2", "problem_latex": "$x^3 - 5x^2 - 2x + 24 = 0$.", "markdown": "$x^3 - 5x^2 - 2x + 24 = 0$.", "answer_latex": [ "$-2$, $3$, $4$." ], "answer_markdown": [ "$-2$, $3$, $4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 5*x**2 - 2*x + 24, 0)", "answer_expr": "[-2, 3, 4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 5*x**2 - 2*x + 24, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page25/3", "set": "dickson-theory-of-equations-1922/ex-page25", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "25", "location": "Exercise Page25, problem 3", "problem_latex": "$x^3 - 10x^2 + 27x - 18 = 0$.", "markdown": "$x^3 - 10x^2 + 27x - 18 = 0$.", "answer_latex": [ "$1$, $3$, $6$." ], "answer_markdown": [ "$1$, $3$, $6$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 10*x**2 + 27*x - 18, 0)", "answer_expr": "[1, 3, 6]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 10*x**2 + 27*x - 18, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page25/4", "set": "dickson-theory-of-equations-1922/ex-page25", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "25", "location": "Exercise Page25, problem 4", "problem_latex": "$x^4 + 4x^3 + 8x + 32 = 0$.", "markdown": "$x^4 + 4x^3 + 8x + 32 = 0$.", "answer_latex": [ "$-2$, $-4$." ], "answer_markdown": [ "$-2$, $-4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 + 4*x**3 + 8*x + 32, 0)", "answer_expr": "[-2, -4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 + 4*x**3 + 8*x + 32, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page25/5", "set": "dickson-theory-of-equations-1922/ex-page25", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "25", "location": "Exercise Page25, problem 5", "problem_latex": "The equation in Ex.~4 of~§23.", "markdown": "The equation in Ex. 4 of §23.", "answer_latex": [ "None." ], "answer_markdown": [ "None." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page27/1", "set": "dickson-theory-of-equations-1922/ex-page27", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "27", "location": "Exercise Page27, problem 1", "problem_latex": "$x^4 - 2x^3 - 21x^2 + 22x + 40 = 0$.", "markdown": "$x^4 - 2x^3 - 21x^2 + 22x + 40 = 0$.", "answer_latex": [ "$2$, $-1$, $-4$, $5$." ], "answer_markdown": [ "$2$, $-1$, $-4$, $5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 2*x**3 - 21*x**2 + 22*x + 40, 0)", "answer_expr": [ 2, -1, -4, 5 ] } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 2*x**3 - 21*x**2 + 22*x + 40, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page19/10" ], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page27/2", "set": "dickson-theory-of-equations-1922/ex-page27", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "27", "location": "Exercise Page27, problem 2", "problem_latex": "$y^3 - 9y^2 - 24y + 216 = 0$.", "markdown": "$y^3 - 9y^2 - 24y + 216 = 0$.", "answer_latex": [ "$9$." ], "answer_markdown": [ "$9$." ], "checks": [ { "task": "solve", "verdict": "FLAG-PARSE", "judge_why": "AttributeError: 'int' object has no attribute 'strip'", "problem_expr": "Eq(y**3 - 9*y**2 - 24*y + 216, 0)", "answer_expr": 9 } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "solve: Eq(x**3 - 9*x**2 - 24*x + 216, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page27/3", "set": "dickson-theory-of-equations-1922/ex-page27", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "27", "location": "Exercise Page27, problem 3", "problem_latex": "$x^4 - 23x^3 + 187x^2 - 653x + 936 = 0$.", "markdown": "$x^4 - 23x^3 + 187x^2 - 653x + 936 = 0$.", "answer_latex": [ "$8$, $9$." ], "answer_markdown": [ "$8$, $9$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 23*x**3 + 187*x**2 - 653*x + 936, 0)", "answer_expr": [ 8, 9 ] } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 23*x**3 + 187*x**2 - 653*x + 936, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page27/4", "set": "dickson-theory-of-equations-1922/ex-page27", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "27", "location": "Exercise Page27, problem 4", "problem_latex": "$x^5 + 47x^4 + 423x^3 + 140x^2 + 1213x - 420 = 0$.", "markdown": "$x^5 + 47x^4 + 423x^3 + 140x^2 + 1213x - 420 = 0$.", "answer_latex": [ "$-12$, $-35$." ], "answer_markdown": [ "$-12$, $-35$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**5 + 47*x**4 + 423*x**3 + 140*x**2 + 1213*x - 420, 0)", "answer_expr": [ -12, -35 ] } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**5 + 47*x**4 + 423*x**3 + 140*x**2 + 1213*x - 420, 0)" ], "shape": [ "solve: Eq(N*x + 3*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page27/5", "set": "dickson-theory-of-equations-1922/ex-page27", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "27", "location": "Exercise Page27, problem 5", "problem_latex": "$x^5 - 34x^3 + 29x^2 + 212x - 300 = 0$.", "markdown": "$x^5 - 34x^3 + 29x^2 + 212x - 300 = 0$.", "answer_latex": [ "$2$, $2$, $-3$." ], "answer_markdown": [ "$2$, $2$, $-3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**5 - 34*x**3 + 29*x**2 + 212*x - 300, 0)", "answer_expr": [ 2, 2, -3 ] } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**5 - 34*x**3 + 29*x**2 + 212*x - 300, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/1", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 1", "problem_latex": "$y^4 -\\frac{40}{3}y^3 + \\frac{130}{3}y^2 - 40y + 9 = 0$.", "markdown": "$y^4 -\\frac{40}{3}y^3 + \\frac{130}{3}y^2 - 40y + 9 = 0$.", "answer_latex": [ "$1$, $3$, $9$, $\\frac{1}{3}$." ], "answer_markdown": [ "$1$, $3$, $9$, $\\frac{1}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**4 - Rational(40,3)*y**3 + Rational(130,3)*y**2 - 40*y + 9, 0)", "answer_expr": "[1, 3, 9, Rational(1,3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 40*x**3/3 + 130*x**2/3 - 40*x + 9, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/10", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 10", "problem_latex": "$y^2 - 2y - \\frac{1}{3} = 0$.", "markdown": "$y^2 - 2y - \\frac{1}{3} = 0$.", "answer_latex": [ "$x^2 - 12x - 12 = 0$." ], "answer_markdown": [ "$x^2 - 12x - 12 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/11", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 11", "problem_latex": "$y^3 - \\frac{1}{2}y^2 - \\frac{1}{3}y + \\frac{1}{4} = 0$.", "markdown": "$y^3 - \\frac{1}{2}y^2 - \\frac{1}{3}y + \\frac{1}{4} = 0$.", "answer_latex": [ "$x^3 - 3x^2 - 12x + 54 = 0$." ], "answer_markdown": [ "$x^3 - 3x^2 - 12x + 54 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/2", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 2", "problem_latex": "$6y^3 - 11y^2 + 6y - 1 = 0$.", "markdown": "$6y^3 - 11y^2 + 6y - 1 = 0$.", "answer_latex": [ "$1$, $\\tfrac{1}{2}$, $\\tfrac{1}{3}$." ], "answer_markdown": [ "$1$, $\\tfrac{1}{2}$, $\\tfrac{1}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(6*y**3 - 11*y**2 + 6*y - 1, 0)", "answer_expr": "[1, Rational(1,2), Rational(1,3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(6*x**3 - 11*x**2 + 6*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N - 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/3", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 3", "problem_latex": "$108y^3 - 270y^2 - 42y + 1 = 0$. [Use $k = 6$.]", "markdown": "$108y^3 - 270y^2 - 42y + 1 = 0$. [Use $k = 6$.]", "answer_latex": [ "$-\\tfrac{1}{6}$." ], "answer_markdown": [ "$-\\tfrac{1}{6}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(108*y**3 - 270*y**2 - 42*y + 1, 0)", "answer_expr": "Rational(-1,6)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(108*x**3 - 270*x**2 - 42*x + 1, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1 +/− ENTER 6 ÷" ], "constants": [ { "value": "1", "source": "the constant term 1 in 108y^3 - 270y^2 - 42y + 1 = 0 (candidate numerator p = ±1 from the rational root test)" }, { "value": "6", "source": "k = 6 in the problem's bracketed hint '[Use k = 6.]' (candidate denominator)" } ], "calculator_value": "-1666666666666666666666666666666667E-34", "printed_value": "-1/6", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page28/4", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 4", "problem_latex": "$32y^3 - 6y - 1 = 0$. [Use the least~$k$.]", "markdown": "$32y^3 - 6y - 1 = 0$. [Use the least $k$.]", "answer_latex": [ "$\\tfrac{1}{2}$, $-\\tfrac{1}{4}$, $-\\tfrac{1}{4}$." ], "answer_markdown": [ "$\\tfrac{1}{2}$, $-\\tfrac{1}{4}$, $-\\tfrac{1}{4}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(32*y**3 - 6*y - 1, 0)", "answer_expr": "[Rational(1,2), Rational(-1,4), Rational(-1,4)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(32*x**3 - 6*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N - 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/5", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 5", "problem_latex": "$96y^3 - 16y^2 - 6y + 1 = 0$.", "markdown": "$96y^3 - 16y^2 - 6y + 1 = 0$.", "answer_latex": [ "$\\tfrac{1}{4}$, $-\\tfrac{1}{4}$, $\\tfrac{1}{6}$." ], "answer_markdown": [ "$\\tfrac{1}{4}$, $-\\tfrac{1}{4}$, $\\tfrac{1}{6}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(96*y**3 - 16*y**2 - 6*y + 1, 0)", "answer_expr": "[Rational(1,4), Rational(-1,4), Rational(1,6)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(96*x**3 - 16*x**2 - 6*x + 1, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/6", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 6", "problem_latex": "$24y^3 - 2y^2 - 5y + 1 = 0$.", "markdown": "$24y^3 - 2y^2 - 5y + 1 = 0$.", "answer_latex": [ "$-\\tfrac{1}{2}$, $\\tfrac{1}{3}$, $\\tfrac{1}{4}$." ], "answer_markdown": [ "$-\\tfrac{1}{2}$, $\\tfrac{1}{3}$, $\\tfrac{1}{4}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(24*y**3 - 2*y**2 - 5*y + 1, 0)", "answer_expr": "[Rational(-1,2), Rational(1,3), Rational(1,4)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(24*x**3 - 2*x**2 - 5*x + 1, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page28/7", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 7", "problem_latex": "$y^3 - \\frac{1}{2}y^2 - 2y + 1 = 0$.", "markdown": "$y^3 - \\frac{1}{2}y^2 - 2y + 1 = 0$.", "answer_latex": [ "$\\tfrac{1}{2}$." ], "answer_markdown": [ "$\\tfrac{1}{2}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**3 - Rational(1,2)*y**2 - 2*y + 1, 0)", "answer_expr": "Rational(1,2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - x**2/2 - 2*x + 1, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1 ENTER 2 ÷", "1 ENTER 2 ÷ −", "1 ENTER 2 ÷ ×", "2 −", "1 ENTER 2 ÷ ×", "1 +", "1 ENTER 2 ÷" ], "constants": [ { "value": "1", "source": "the 1 in the printed constant term '+ 1' and in the '1/2' of '- 1/2 y^2'" }, { "value": "2", "source": "the 2 in the denominator of '1/2' and the 2 in '- 2y'" } ], "calculator_value": "+5E-1", "printed_value": "1/2", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page28/8", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 8", "problem_latex": "$y^3 - \\frac{2}{3}y^2 + 3y - 2 = 0$.", "markdown": "$y^3 - \\frac{2}{3}y^2 + 3y - 2 = 0$.", "answer_latex": [ "$\\tfrac{2}{3}$." ], "answer_markdown": [ "$\\tfrac{2}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**3 - Rational(2,3)*y**2 + 3*y - 2, 0)", "answer_expr": "Rational(2,3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 2*x**2/3 + 3*x - 2, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 3 ÷ ENTER 2 ENTER 3 ÷ −", "× 3 +", "2 ENTER 3 ÷ × 2 −", "2 ENTER 3 ÷" ], "constants": [ { "value": "2", "source": "the 2 in the constant term −2, and the numerator of the 2/3 in the problem" }, { "value": "3", "source": "the denominator of the 2/3 in the problem, and the coefficient 3 of y in 3y" } ], "calculator_value": "+6666666666666666666666666666666667E-34", "printed_value": "2/3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page28/9", "set": "dickson-theory-of-equations-1922/ex-page28", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "28", "location": "Exercise Page28, problem 9", "problem_latex": "Solve Exs.~2--6 by replacing $y$ by~$1/x$.", "markdown": "Solve Exs. 2--6 by replacing $y$ by $1/x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page30/1", "set": "dickson-theory-of-equations-1922/ex-page30", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "30", "location": "Exercise Page30, problem 1", "problem_latex": "$x^2 - 5x + 4 = 0$.", "markdown": "$x^2 - 5x + 4 = 0$.", "answer_latex": [ "$1$, $4$." ], "answer_markdown": [ "$1$, $4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 5*x + 4, 0)", "answer_expr": "[1, 4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 5*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page30/2", "set": "dickson-theory-of-equations-1922/ex-page30", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "30", "location": "Exercise Page30, problem 2", "problem_latex": "$x^2 + 5x + 4 = 0$.", "markdown": "$x^2 + 5x + 4 = 0$.", "answer_latex": [ "$-1$, $-4$." ], "answer_markdown": [ "$-1$, $-4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 + 5*x + 4, 0)", "answer_expr": "[-1, -4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 + 5*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page30/3", "set": "dickson-theory-of-equations-1922/ex-page30", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "30", "location": "Exercise Page30, problem 3", "problem_latex": "$x^2 + 5x - 4 = 0$.", "markdown": "$x^2 + 5x - 4 = 0$.", "answer_latex": [ "$0.7$, $-5.7$." ], "answer_markdown": [ "$0.7$, $-5.7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 + 5*x - 4, 0)", "answer_expr": "[0.7, -5.7]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 + 5*x - 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page30/4", "set": "dickson-theory-of-equations-1922/ex-page30", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "30", "location": "Exercise Page30, problem 4", "problem_latex": "$x^2 - 5x - 4 = 0$.", "markdown": "$x^2 - 5x - 4 = 0$.", "answer_latex": [ "$-0.7$, $5.7$." ], "answer_markdown": [ "$-0.7$, $5.7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 5*x - 4, 0)", "answer_expr": "[-0.7, 5.7]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 5*x - 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page30/5", "set": "dickson-theory-of-equations-1922/ex-page30", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "30", "location": "Exercise Page30, problem 5", "problem_latex": "$x^2 - 4x + 4 = 0$.", "markdown": "$x^2 - 4x + 4 = 0$.", "answer_latex": [ "$2$, $2$." ], "answer_markdown": [ "$2$, $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 4*x + 4, 0)", "answer_expr": "[2, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 4*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page30/6", "set": "dickson-theory-of-equations-1922/ex-page30", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "30", "location": "Exercise Page30, problem 6", "problem_latex": "$x^2 - 3x + 4 = 0$.", "markdown": "$x^2 - 3x + 4 = 0$.", "answer_latex": [ "Imaginary." ], "answer_markdown": [ "Imaginary." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 3*x + 4, 0)", "answer_expr": "[]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 3*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/1", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 1", "problem_latex": "Show by~\\Eq{16} that the roots of~\\Eq{12} are $2\\cos 2\\pi/7$, $2\\cos 4\\pi/7$, $2\\cos 6\\pi/7$.", "markdown": "Show by $(16)$ that the roots of $(12)$ are $2\\cos 2\\pi/7$, $2\\cos 4\\pi/7$, $2\\cos 6\\pi/7$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/10", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 10", "problem_latex": "To construct a straight line representing the distance from the circular base\nof a hemisphere to the parallel plane which bisects the hemisphere.", "markdown": "To construct a straight line representing the distance from the circular base of a hemisphere to the parallel plane which bisects the hemisphere.", "answer_latex": [ "See~\\Eq{11},~§32." ], "answer_markdown": [ "See $(11)$, §32." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:ruler_compass_impossibility" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/11", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 11", "problem_latex": "To construct lines representing the lengths of the edges of an existing rectangular\nparallelopiped having a diagonal of length~$5$, surface area~$24$, and volume~$1$, $2$, $3$, or~$5$.", "markdown": "To construct lines representing the lengths of the edges of an existing rectangular parallelopiped having a diagonal of length $5$, surface area $24$, and volume $1$, $2$, $3$, or $5$.", "answer_latex": [ "Edges roots of $x^3 - 7x^2 + 12x - v = 0$, all real~(§45) and irrational." ], "answer_markdown": [ "Edges roots of $x^3 - 7x^2 + 12x - v = 0$, all real (§45) and irrational." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "other:ruler_compass_impossibility" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/12", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 12", "problem_latex": "To trisect an angle whose cosine is $\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{1}{4}$, $\\frac{1}{8}$ or~$p/q$, where $p$ and~$q$ ($q>1$) are\nintegers without a common factor, and $q$ is not divisible by a cube.", "markdown": "To trisect an angle whose cosine is $\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{1}{4}$, $\\frac{1}{8}$ or $p/q$, where $p$ and $q$ ($q>1$) are integers without a common factor, and $q$ is not divisible by a cube.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:ruler_compass_impossibility" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/13", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 13, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 13", "problem_latex": "To trisect an angle whose cosine is $(4a^3 - 3ab^2)/b^3$, where the integer~$a$ is numerically\nless than the integer~$b$; for example, $\\cos^{-1} 11/16$ if $a = -1$, $b = 4$.", "markdown": "To trisect an angle whose cosine is $(4a^3 - 3ab^2)/b^3$, where the integer $a$ is numerically less than the integer $b$; for example, $\\cos^{-1} 11/16$ if $a = -1$, $b = 4$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.subst", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/14", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 14", "problem_latex": "To construct the legs of a right triangle, given its area and hypotenuse.", "markdown": "To construct the legs of a right triangle, given its area and hypotenuse.", "answer_latex": [ "$\\Delta = \\text{area}$, $c = \\text{hypotenuse}$, squares of legs $\\tfrac{1}{2}(c^2 ± \\sqrt{c^4 - 16\\Delta^2})$." ], "answer_markdown": [ "$\\Delta = \\text{area}$, $c = \\text{hypotenuse}$, squares of legs $\\tfrac{1}{2}(c^2 ± \\sqrt{c^4 - 16\\Delta^2})$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "other:ruler_compass_construction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/15", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 15, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 15", "problem_latex": "To construct the third side of a triangle, given two sides and its area.", "markdown": "To construct the third side of a triangle, given two sides and its area.", "answer_latex": [ "$\\Delta$ area, $a$, $b$ given sides, square third side is $a^2 + b^2 ± 2\\sqrt{a^2b^2 - 4\\Delta^2}$." ], "answer_markdown": [ "$\\Delta$ area, $a$, $b$ given sides, square third side is $a^2 + b^2 ± 2\\sqrt{a^2b^2 - 4\\Delta^2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.eqn", "core.trig", "other:ruler_compass_construction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/16", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 16, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 16", "problem_latex": "To locate the point~$P$ on the side $BC=1$ of a given square $ABCD$ such that\nthe straight line $AP$ cuts $DC$ produced at a point~$Q$ for which the\nlength of $PQ$ is a given\nnumber~$g$. Show that $y=BP$ is a root of a reciprocal quartic equation, and solve it\nwhen $g = 10$.", "markdown": "To locate the point $P$ on the side $BC=1$ of a given square $ABCD$ such that the straight line $AP$ cuts $DC$ produced at a point $Q$ for which the length of $PQ$ is a given number $g$. Show that $y=BP$ is a root of a reciprocal quartic equation, and solve it when $g = 10$.", "answer_latex": [ "$y^4 - 2y^3 + (2 - g^2)y^2 - 2y + 1 = 0$, pos.\\ roots $0.09125$, $10.95862$." ], "answer_markdown": [ "$y^4 - 2y^3 + (2 - g^2)y^2 - 2y + 1 = 0$, pos. roots $0.09125$, $10.95862$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/2", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 2", "problem_latex": "The imaginary fifth roots of unity satisfy\n$y^4 + y^3 + y^2 + y + 1 = 0$, which by the substitution~\\Eq{14} becomes\n$x^2 + x - 1 = 0$. It has the root\n\\[\nR + \\frac{1}{R} = 2 \\cos\\frac{2\\pi}{5} = \\frac{1}{2}(\\sqrt{5}-1).\n\\]\nIn a circle of radius unity and center~$O$ draw two perpendicular\ndiameters $AOA'$, $BOB'$. With the middle\npoint~$M$ of~$OA'$ as center and radius~$MB$ draw a circle\ncutting~$OA$ at~$C$ (Fig.~10). Show that $OC$ and~$BC$\nare the sides~$s_{10}$ and~$s_5$ of the inscribed regular decagon\nand pentagon respectively. Hints:", "markdown": "The imaginary fifth roots of unity satisfy $y^4 + y^3 + y^2 + y + 1 = 0$, which by the substitution $(14)$ becomes $x^2 + x - 1 = 0$. It has the root R + 1R = 2 25 = 12(5-1). In a circle of radius unity and center $O$ draw two perpendicular diameters $AOA'$, $BOB'$. With the middle point $M$ of $OA'$ as center and radius $MB$ draw a circle cutting $OA$ at $C$ (Fig. 10). Show that $OC$ and $BC$ are the sides $s_{10}$ and $s_5$ of the inscribed regular decagon and pentagon respectively. Hints:", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:ruler_compass_construction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/3", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 3", "problem_latex": "If $R$ is a root of~\\Eq{19} verify as at the end of~§35 that $R+R^8$, $R^2+R^7$, and $R^4+R^5$\nare the roots of~\\Eq{11}.", "markdown": "If $R$ is a root of $(19)$ verify as at the end of §35 that $R+R^8$, $R^2+R^7$, and $R^4+R^5$ are the roots of $(11)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/4", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 4", "problem_latex": "Hence show that the roots of~\\Eq{11} are $2\\cos 2\\pi/9$, $2\\cos 4\\pi/9$, $2\\cos 8\\pi/9$.", "markdown": "Hence show that the roots of $(11)$ are $2\\cos 2\\pi/9$, $2\\cos 4\\pi/9$, $2\\cos 8\\pi/9$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/5", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 5", "problem_latex": "Reduce $y^{11} = 1$ to an equation of degree~$5$ in~$x$.", "markdown": "Reduce $y^{11} = 1$ to an equation of degree $5$ in $x$.", "answer_latex": [ "$x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1 = 0$." ], "answer_markdown": [ "$x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/6", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 6", "problem_latex": "Solve $y^5 - 7y^4 + y^3 - y^2 + 7y - 1 = 0$ by radicals. [One root is~$1$.]", "markdown": "Solve $y^5 - 7y^4 + y^3 - y^2 + 7y - 1 = 0$ by radicals. [One root is $1$.]", "answer_latex": [ "$-\\tfrac{1}{2}(1±\\sqrt{-3})$, $\\tfrac{1}{2}(7±\\sqrt{45})$." ], "answer_markdown": [ "$-\\tfrac{1}{2}(1±\\sqrt{-3})$, $\\tfrac{1}{2}(7±\\sqrt{45})$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**5 - 7*y**4 + y**3 - y**2 + 7*y - 1, 0)", "answer_expr": [ "-1/2 + sqrt(-3)/2", "-1/2 - sqrt(-3)/2", "(7 + sqrt(45))/2", "(7 - sqrt(45))/2" ] } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**5 - 7*x**4 + x**3 - x**2 + 7*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + x**N - 1, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.complex", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/7", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 7", "problem_latex": "After finding so easily in \\ChapRef{I} the trigonometric forms of the complex roots\nof unity, why do we now go to so much additional trouble to find them algebraically?", "markdown": "After finding so easily in [chap:I]Chapter I the trigonometric forms of the complex roots of unity, why do we now go to so much additional trouble to find them algebraically?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/8", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 8", "problem_latex": "Prove that every real root of $x^4 + ax^2 + b = 0$ can be constructed with ruler and\ncompasses, given lines of lengths $a$ and~$b$.", "markdown": "Prove that every real root of $x^4 + ax^2 + b = 0$ can be constructed with ruler and compasses, given lines of lengths $a$ and $b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:ruler_compass_construction" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page40/9", "set": "dickson-theory-of-equations-1922/ex-page40", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "39", "location": "Exercise Page40, problem 9", "problem_latex": "Show that the real roots of $x^3 - px - q = 0$ are the abscissas of the intersections\nof the parabola $y = x^2$ and the circle through the origin with the center\n$(\\frac{1}{2}q, \\frac{1}{2} + \\frac{1}{2}p)$.", "markdown": "Show that the real roots of $x^3 - px - q = 0$ are the abscissas of the intersections of the parabola $y = x^2$ and the circle through the origin with the center $(\\frac{1}{2}q, \\frac{1}{2} + \\frac{1}{2}p)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.factor", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/1", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 1", "problem_latex": "If $a$ and~$b$ are relatively prime numbers, so that their greatest common divisor\nis unity, we can find integers $c$ and~$d$ such that $ac + bd = 1$. Show that, if regular polygons\nof $a$ and~$b$ sides can be constructed and hence angles $2\\pi/a$ and $2\\pi/b$, a regular\npolygon of $a·b$ sides can be derived.", "markdown": "If $a$ and $b$ are relatively prime numbers, so that their greatest common divisor is unity, we can find integers $c$ and $d$ such that $ac + bd = 1$. Show that, if regular polygons of $a$ and $b$ sides can be constructed and hence angles $2\\pi/a$ and $2\\pi/b$, a regular polygon of $a·b$ sides can be derived.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/2", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 2", "problem_latex": "If $p = 2^h + 1$ is a prime, $h$ is a power of~$2$. For $h = 2^0$, $2^1$, $2^2$, $2^3$, the values of~$p$\nare $3$, $5$, $17$, $257$ and are primes. [Show that $h$ cannot have an odd factor other than\nunity.]", "markdown": "If $p = 2^h + 1$ is a prime, $h$ is a power of $2$. For $h = 2^0$, $2^1$, $2^2$, $2^3$, the values of $p$ are $3$, $5$, $17$, $257$ and are primes. [Show that $h$ cannot have an odd factor other than unity.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/3", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 3", "problem_latex": "For $13$th roots of unity find the least~$g$~(§38), write out the three periods each\nof four terms, and find the cubic equation having them as roots.", "markdown": "For $13$th roots of unity find the least $g$ (§38), write out the three periods each of four terms, and find the cubic equation having them as roots.", "answer_latex": [ "$g=2$, $R + R^8 + R^{12} + R^5$, etc., $z^3 + z^2 - 4z + 1 = 0$." ], "answer_markdown": [ "$g=2$, $R + R^8 + R^{12} + R^5$, etc., $z^3 + z^2 - 4z + 1 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.arith", "other:gauss_periods" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/4", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 4", "problem_latex": "For the primitive ninth roots of unity find the least~$g$ and write out the three\nperiods each of two terms.", "markdown": "For the primitive ninth roots of unity find the least $g$ and write out the three periods each of two terms.", "answer_latex": [ "$g=2$, $R+R^8$, $R^2+R^7$, $R^4+R^5$." ], "answer_markdown": [ "$g=2$, $R+R^8$, $R^2+R^7$, $R^4+R^5$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "other:gauss_periods" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/5", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 5", "problem_latex": "$y^4 + 4y^3 - 3y^2 + 4y + 1 = 0$.", "markdown": "$y^4 + 4y^3 - 3y^2 + 4y + 1 = 0$.", "answer_latex": [ "$\\tfrac{1}{2}(1±\\sqrt{-3})$, $\\tfrac{1}{2}(-5±\\sqrt{21})$." ], "answer_markdown": [ "$\\tfrac{1}{2}(1±\\sqrt{-3})$, $\\tfrac{1}{2}(-5±\\sqrt{21})$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**4 + 4*y**3 - 3*y**2 + 4*y + 1, 0)", "answer_expr": "[(1+sqrt(-3))/2, (1-sqrt(-3))/2, (-5+sqrt(21))/2, (-5-sqrt(21))/2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 + 4*x**3 - 3*x**2 + 4*x + 1, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/6", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 6", "problem_latex": "$y^5 - 4y^4 + y^3 + y^2 - 4y + 1 = 0$.", "markdown": "$y^5 - 4y^4 + y^3 + y^2 - 4y + 1 = 0$.", "answer_latex": [ "$-1$, $2±\\sqrt{3}$, $\\tfrac{1}{2} ± \\tfrac{1}{2}\\sqrt{-3}$." ], "answer_markdown": [ "$-1$, $2±\\sqrt{3}$, $\\tfrac{1}{2} ± \\tfrac{1}{2}\\sqrt{-3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**5 - 4*y**4 + y**3 + y**2 - 4*y + 1, 0)", "answer_expr": "[-1, 2+sqrt(3), 2-sqrt(3), (1+sqrt(-3))/2, (1-sqrt(-3))/2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**5 - 4*x**4 + x**3 + x**2 - 4*x + 1, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + 3*x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/7", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 7", "problem_latex": "$2y^6 - 5y^5 + 4y^4 - 4y^2 + 5y - 2 = 0$.", "markdown": "$2y^6 - 5y^5 + 4y^4 - 4y^2 + 5y - 2 = 0$.", "answer_latex": [ "$1$, $1$, $1$, $-1$, $\\tfrac{1}{4}(1±\\sqrt{-15})$." ], "answer_markdown": [ "$1$, $1$, $1$, $-1$, $\\tfrac{1}{4}(1±\\sqrt{-15})$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2*y**6 - 5*y**5 + 4*y**4 - 4*y**2 + 5*y - 2, 0)", "answer_expr": "[1, 1, 1, -1, (1+sqrt(-15))/4, (1-sqrt(-15))/4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x**6 - 5*x**5 + 4*x**4 - 4*x**2 + 5*x - 2, 0)" ], "shape": [ "solve: Eq(N*x + 4*N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page44/8", "set": "dickson-theory-of-equations-1922/ex-page44", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "44", "location": "Exercise Page44, problem 8", "problem_latex": "$y^5 + 1 = 31(y + 1)^5$.", "markdown": "$y^5 + 1 = 31(y + 1)^5$.", "answer_latex": [ "$-1$, $-2$, $-\\tfrac{1}{2}$, $\\tfrac{1}{6}(-5±\\sqrt{-11})$." ], "answer_markdown": [ "$-1$, $-2$, $-\\tfrac{1}{2}$, $\\tfrac{1}{6}(-5±\\sqrt{-11})$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**5 + 1, 31*(y + 1)**5)", "answer_expr": "[-1, -2, -1/2, (-5+sqrt(-11))/6, (-5-sqrt(-11))/6]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**5 + 1, 31*(x + 1)**5)" ], "shape": [ "solve: Eq(x**N + 1, N*(x + 1)**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page46/1", "set": "dickson-theory-of-equations-1922/ex-page46", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Exercise Page46, problem 1", "problem_latex": "$y^3 - 18y + 35 = 0$.", "markdown": "$y^3 - 18y + 35 = 0$.", "answer_latex": [ "$-5$, $\\tfrac{1}{2}(5±\\sqrt{-3})$." ], "answer_markdown": [ "$-5$, $\\tfrac{1}{2}(5±\\sqrt{-3})$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**3 - 18*y + 35, 0)", "answer_expr": "[-5, (5 + sqrt(-3))/2, (5 - sqrt(-3))/2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 18*x + 35, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page46/2", "set": "dickson-theory-of-equations-1922/ex-page46", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Exercise Page46, problem 2", "problem_latex": "$x^3 + 6x^2 + 3x + 18 = 0$.", "markdown": "$x^3 + 6x^2 + 3x + 18 = 0$.", "answer_latex": [ "$-6$, $±\\sqrt{-3}$." ], "answer_markdown": [ "$-6$, $±\\sqrt{-3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 6*x**2 + 3*x + 18, 0)", "answer_expr": "[-6, sqrt(-3), -sqrt(-3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 6*x**2 + 3*x + 18, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page46/3", "set": "dickson-theory-of-equations-1922/ex-page46", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Exercise Page46, problem 3", "problem_latex": "$y^3 - 2y + 4 = 0$.", "markdown": "$y^3 - 2y + 4 = 0$.", "answer_latex": [ "$-2$, $1± i$." ], "answer_markdown": [ "$-2$, $1± i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(y**3 - 2*y + 4, 0) fails at {y: I + 1}: leaves -2.0*I + (I + 1.0)**3 + 2.0", "problem_expr": "Eq(y**3 - 2*y + 4, 0)", "answer_expr": "[-2, 1 + I, 1 - I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**3 - 2*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page46/4", "set": "dickson-theory-of-equations-1922/ex-page46", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "46", "location": "Exercise Page46, problem 4", "problem_latex": "$28x^3 + 9x^2 - 1 = 0$.", "markdown": "$28x^3 + 9x^2 - 1 = 0$.", "answer_latex": [ "$\\tfrac{1}{4}$, $\\tfrac{1}{7}(-2±\\sqrt{-3})$." ], "answer_markdown": [ "$\\tfrac{1}{4}$, $\\tfrac{1}{7}(-2±\\sqrt{-3})$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(28*x**3 + 9*x**2 - 1, 0)", "answer_expr": "[Rational(1, 4), (-2 + sqrt(-3))/7, (-2 - sqrt(-3))/7]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(28*x**3 + 9*x**2 - 1, 0)" ], "shape": [ "solve: Eq(2*N*x**N - 1, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page99/2" ], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page48/1", "set": "dickson-theory-of-equations-1922/ex-page48", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Exercise Page48, problem 1", "problem_latex": "$y^3 - 2y - 4 = 0$.", "markdown": "$y^3 - 2y - 4 = 0$.", "answer_latex": [ "$\\Delta = -400$, one." ], "answer_markdown": [ "$\\Delta = -400$, one." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -400.0, printed -400 (half-unit 0.5; correctly rounded at the printed digits: -400.0)", "problem_expr": "discriminant(y**3 - 2*y - 4, y)", "answer_expr": "-400" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -400" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X=-400", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page48/2", "set": "dickson-theory-of-equations-1922/ex-page48", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Exercise Page48, problem 2", "problem_latex": "$y^3 - 15y + 4 = 0$.", "markdown": "$y^3 - 15y + 4 = 0$.", "answer_latex": [ "$\\Delta = 4 · 27 · 121$, three." ], "answer_markdown": [ "$\\Delta = 4 · 27 · 121$, three." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 13068.0, printed 13068 (half-unit 0.5; correctly rounded at the printed digits: 13068.0)", "problem_expr": "discriminant(y**3 - 15*y + 4, y)", "answer_expr": "4*27*121" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 13068" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page48/3", "set": "dickson-theory-of-equations-1922/ex-page48", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Exercise Page48, problem 3", "problem_latex": "$y^3 - 27y + 54 = 0$.", "markdown": "$y^3 - 27y + 54 = 0$.", "answer_latex": [ "$\\Delta = 0$, two." ], "answer_markdown": [ "$\\Delta = 0$, two." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.0, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)", "problem_expr": "discriminant(y**3 - 27*y + 54, y)", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 0" ], "shape": [ "evaluate: 0" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page48/4" ], "needs": [ "core.arith" ], "expectation": "X=0", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page48/4", "set": "dickson-theory-of-equations-1922/ex-page48", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Exercise Page48, problem 4", "problem_latex": "$x^3 + 4x^2 - 11x + 6 = 0$.", "markdown": "$x^3 + 4x^2 - 11x + 6 = 0$.", "answer_latex": [ "$\\Delta = 0$, two." ], "answer_markdown": [ "$\\Delta = 0$, two." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.0, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)", "problem_expr": "discriminant(x**3 + 4*x**2 - 11*x + 6, x)", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 0" ], "shape": [ "evaluate: 0" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page48/3" ], "needs": [ "core.arith" ], "expectation": "X=0", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page48/5", "set": "dickson-theory-of-equations-1922/ex-page48", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "48", "location": "Exercise Page48, problem 5", "problem_latex": "Show by means of~§21 that a double root of a real cubic is real.", "markdown": "Show by means of §21 that a double root of a real cubic is real.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page49/1", "set": "dickson-theory-of-equations-1922/ex-page49", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Exercise Page49, problem 1", "problem_latex": "Solve $y^3 -15y+4=0$.", "markdown": "Solve $y^3 -15y+4=0$.", "answer_latex": [ "$-4$, $2±\\sqrt{3}$." ], "answer_markdown": [ "$-4$, $2±\\sqrt{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**3 - 15*y + 4, 0)", "answer_expr": "[-4, 2 + sqrt(3), 2 - sqrt(3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 15*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page49/2", "set": "dickson-theory-of-equations-1922/ex-page49", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Exercise Page49, problem 2", "problem_latex": "Solve $y^3 -2y-1=0$.", "markdown": "Solve $y^3 -2y-1=0$.", "answer_latex": [ "See Ex.~1,~§47." ], "answer_markdown": [ "See Ex. 1, §47." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(y**3 - 2*y - 1, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**3 - 2*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + x**N - 1, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page49/3", "set": "dickson-theory-of-equations-1922/ex-page49", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Exercise Page49, problem 3", "problem_latex": "Solve $y^3 -7y+7=0$.", "markdown": "Solve $y^3 -7y+7=0$.", "answer_latex": [ "$1.3569$, $1.6920$, $-3.0489$." ], "answer_markdown": [ "$1.3569$, $1.6920$, $-3.0489$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(y**3 - 7*y + 7, 0)", "answer_expr": "[1.3569, 1.6920, -3.0489]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 7*x + 7, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page49/4", "set": "dickson-theory-of-equations-1922/ex-page49", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Exercise Page49, problem 4", "problem_latex": "Solve $x^3+ 3x^2 -2x-5=0$.", "markdown": "Solve $x^3+ 3x^2 -2x-5=0$.", "answer_latex": [ "$-1.201639$, $1.330058$, $-3.128419$." ], "answer_markdown": [ "$-1.201639$, $1.330058$, $-3.128419$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 3*x**2 - 2*x - 5, 0)", "answer_expr": "[-1.201639, 1.330058, -3.128419]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 3*x**2 - 2*x - 5, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page89/2" ], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page49/5", "set": "dickson-theory-of-equations-1922/ex-page49", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Exercise Page49, problem 5", "problem_latex": "Solve $x^3 +x^2 -2x-1=0$.", "markdown": "Solve $x^3 +x^2 -2x-1=0$.", "answer_latex": [ "$1.24698$, $-1.80194$, $-0.44504$." ], "answer_markdown": [ "$1.24698$, $-1.80194$, $-0.44504$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + x**2 - 2*x - 1, 0)", "answer_expr": "[1.24698, -1.80194, -0.44504]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + x**2 - 2*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + 2*x**N - 1, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page89/3" ], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page49/6", "set": "dickson-theory-of-equations-1922/ex-page49", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "49", "location": "Exercise Page49, problem 6", "problem_latex": "Solve $x^3 +4x^2 -7=0$.", "markdown": "Solve $x^3 +4x^2 -7=0$.", "answer_latex": [ "$1.1642$, $-1.7729$, $-3.3914$." ], "answer_markdown": [ "$1.1642$, $-1.7729$, $-3.3914$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 4*x**2 - 7, 0)", "answer_expr": "[1.1642, -1.7729, -3.3914]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 4*x**2 - 7, 0)" ], "shape": [ "solve: Eq(N*x**N + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page89/7" ], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page51/1", "set": "dickson-theory-of-equations-1922/ex-page51", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Exercise Page51, problem 1", "problem_latex": "Solve $x^4 - 8x^3 + 9x^2 + 8x - 10 = 0$. Note that~\\Eq{17} is $(y - 9) (y^2 - 24) = 0$.", "markdown": "Solve $x^4 - 8x^3 + 9x^2 + 8x - 10 = 0$. Note that $(17)$ is $(y - 9) (y^2 - 24) = 0$.", "answer_latex": [ "$1$, $-1$, $4±\\sqrt{6}$." ], "answer_markdown": [ "$1$, $-1$, $4±\\sqrt{6}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 8*x**3 + 9*x**2 + 8*x - 10, 0)", "answer_expr": "[1, -1, 4 + sqrt(6), 4 - sqrt(6)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 8*x**3 + 9*x**2 + 8*x - 10, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page51/2", "set": "dickson-theory-of-equations-1922/ex-page51", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Exercise Page51, problem 2", "problem_latex": "Solve $x^4 - 2x^3 - 7x^2 + 8x + 12 = 0$. Since the right member of~\\Eq{16} is\n$(8 + y) (x^2 - x) + \\frac{1}{4} y^2 - 12$, use $y = -8$.", "markdown": "Solve $x^4 - 2x^3 - 7x^2 + 8x + 12 = 0$. Since the right member of $(16)$ is $(8 + y) (x^2 - x) + \\frac{1}{4} y^2 - 12$, use $y = -8$.", "answer_latex": [ "$-1$, $-2$, $2$, $3$." ], "answer_markdown": [ "$-1$, $-2$, $2$, $3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 2*x**3 - 7*x**2 + 8*x + 12, 0)", "answer_expr": "[-1, -2, 2, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 2*x**3 - 7*x**2 + 8*x + 12, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page51/3", "set": "dickson-theory-of-equations-1922/ex-page51", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Exercise Page51, problem 3", "problem_latex": "Solve $x^4 - 3x^2 + 6x - 2 = 0$.", "markdown": "Solve $x^4 - 3x^2 + 6x - 2 = 0$.", "answer_latex": [ "$1± i$, $-1±\\sqrt{2}$." ], "answer_markdown": [ "$1± i$, $-1±\\sqrt{2}$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**4 - 3*x**2 + 6*x - 2, 0) fails at {x: I + 1}: leaves 6.0*I + (I + 1.0)**4 - 3.0*(I + 1.0)**2 + 4.0", "problem_expr": "Eq(x**4 - 3*x**2 + 6*x - 2, 0)", "answer_expr": "[1 + I, 1 - I, -1 + sqrt(2), -1 - sqrt(2)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**4 - 3*x**2 + 6*x - 2, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page51/4", "set": "dickson-theory-of-equations-1922/ex-page51", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Exercise Page51, problem 4", "problem_latex": "Solve $x^4 - 2x^2 - 8x - 3 = 0$.", "markdown": "Solve $x^4 - 2x^2 - 8x - 3 = 0$.", "answer_latex": [ "$1±\\sqrt{2}$, $-1±\\sqrt{-2}$." ], "answer_markdown": [ "$1±\\sqrt{2}$, $-1±\\sqrt{-2}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 2*x**2 - 8*x - 3, 0)", "answer_expr": "[1 + sqrt(2), 1 - sqrt(2), -1 + sqrt(-2), -1 - sqrt(-2)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 2*x**2 - 8*x - 3, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page51/5", "set": "dickson-theory-of-equations-1922/ex-page51", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "51", "location": "Exercise Page51, problem 5", "problem_latex": "Solve $x^4 - 10x^2 - 20x - 16 = 0$.", "markdown": "Solve $x^4 - 10x^2 - 20x - 16 = 0$.", "answer_latex": [ "$4$, $-2$, $-1± i$." ], "answer_markdown": [ "$4$, $-2$, $-1± i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**4 - 10*x**2 - 20*x - 16, 0) fails at {x: I - 1}: leaves -20.0*I + (I - 1.0)**4 - 10.0*(I - 1.0)**2 + 4.0", "problem_expr": "Eq(x**4 - 10*x**2 - 20*x - 16, 0)", "answer_expr": "[4, -2, -1 + I, -1 - I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**4 - 10*x**2 - 20*x - 16, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/1", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 1", "problem_latex": "Find the coordinates of the single real point of intersection of the parabola\n$y = x^2$ and the hyperbola $xy - 4x + y + 6 = 0$.", "markdown": "Find the coordinates of the single real point of intersection of the parabola $y = x^2$ and the hyperbola $xy - 4x + y + 6 = 0$.", "answer_latex": [ "$(-3, 9)$." ], "answer_markdown": [ "$(-3, 9)$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: -3, y: 9}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, x**2), Eq(a*x + a - 4*x + 6, 0))" ], "shape": [ "solve: (Eq(a, x**N), Eq(N*x + N + a*x + a, 0))" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/2", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 2", "problem_latex": "Show that the abscissas of the points of intersection of $y=x^2$ and\n$ax^2 - xy + y^2 - x - (a+5)y - 6 = 0$\nare the roots of $x^4 - x^3 - 5x^2 - x - 6 = 0$. Compute the discriminant\nof the latter and show that only two of the four points of intersection are real.", "markdown": "Show that the abscissas of the points of intersection of $y=x^2$ and $ax^2 - xy + y^2 - x - (a+5)y - 6 = 0$ are the roots of $x^4 - x^3 - 5x^2 - x - 6 = 0$. Compute the discriminant of the latter and show that only two of the four points of intersection are real.", "answer_latex": [ "$\\Delta=-250000$, $x=3$, $-2$, $±i$." ], "answer_markdown": [ "$\\Delta=-250000$, $x=3$, $-2$, $±i$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "other:discriminant" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/3", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 3", "problem_latex": "Find the coordinates of the two real points in Ex.~2.", "markdown": "Find the coordinates of the two real points in Ex. 2.", "answer_latex": [ "$(3,9)$, $(-2,4)$." ], "answer_markdown": [ "$(3,9)$, $(-2,4)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/4", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 4", "problem_latex": "A right prism of height~$h$ has a square base whose side is~$b$ and whose diagonal\nis therefore $b\\sqrt{2}$. If $v$ denotes the volume and $d$ a diagonal of the prism, $v = hb^2$ and\n$d^2 = h^2 + (b\\sqrt{2})^2$. Multiply the last equation by~$h$ and replace $hb^2$ by~$v$. Hence\n$h^3 - d^2h + 2v = 0$.\nIts discriminant is zero if $d = 3\\sqrt{3}$, $v = 27$; find~$h$.", "markdown": "A right prism of height $h$ has a square base whose side is $b$ and whose diagonal is therefore $b\\sqrt{2}$. If $v$ denotes the volume and $d$ a diagonal of the prism, $v = hb^2$ and $d^2 = h^2 + (b\\sqrt{2})^2$. Multiply the last equation by $h$ and replace $hb^2$ by $v$. Hence $h^3 - d^2h + 2v = 0$. Its discriminant is zero if $d = 3\\sqrt{3}$, $v = 27$; find $h$.", "answer_latex": [ "$h=3$." ], "answer_markdown": [ "$h=3$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(h**3 - (3*sqrt(3))**2*h + 2*27, 0)", "answer_expr": "3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x**3 - 27*x + 54, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": "X=3", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/5", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 5", "problem_latex": "Find the admissible values of~$h$ in Ex.~4 when $d = 12$, $v = 332.5$.", "markdown": "Find the admissible values of $h$ in Ex. 4 when $d = 12$, $v = 332.5$.", "answer_latex": [ "$6.856$, $7$." ], "answer_markdown": [ "$6.856$, $7$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(h**3 - 144*h + 665, 0)", "answer_expr": "[6.856, 7]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x**3 - 144*x + 665, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/6", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 6", "problem_latex": "Find a necessary and sufficient condition that quartic equation~\\Eq{15} shall have\none root the negative of another root.\n\nHint: $(x_1 + x_2)(x_3 + x_4) = q - y_1$. Hence substitute $q$ for~$y$ in~\\Eq{17}.", "markdown": "Find a necessary and sufficient condition that quartic equation $(15)$ shall have one root the negative of another root. Hint: $(x_1 + x_2)(x_3 + x_4) = q - y_1$. Hence substitute $q$ for $y$ in $(17)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "other:quartic_resolvent" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/7", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 7", "problem_latex": "In the study of parabolic orbits occurs the equation\n\\[% [** PP: Displayed for better line breaking.]\n\\tan\\tfrac{1}{2}v + \\tfrac{1}{3}\\tan^3 \\tfrac{1}{2}v = t.\n\\]\nProve that there is a single real root and that it has the same sign as~$t$.", "markdown": "In the study of parabolic orbits occurs the equation % [** PP: Displayed for better line breaking.] 12v + 13^3 12v = t. Prove that there is a single real root and that it has the same sign as $t$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page54/8", "set": "dickson-theory-of-equations-1922/ex-page54", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "54", "location": "Exercise Page54, problem 8", "problem_latex": "In the problem of three astronomical bodies occurs the equation $x^3 + ax + 2 = 0$.\nProve that it has three real roots if and only if $a\\leqq{-3}$.", "markdown": "In the problem of three astronomical bodies occurs the equation $x^3 + ax + 2 = 0$. Prove that it has three real roots if and only if $a\\leqq{-3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:discriminant" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/1", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 1", "problem_latex": "Show that the slope of the tangent to $y = 8x^3 - 22x^2 + 13x - 2$ at $(x, y)$ is\n$24x^2 - 44x + 13$, and that the bend points are $(0.37, 0.203)$, $(1.46, -5.03)$, approximately.\nDraw the graph.", "markdown": "Show that the slope of the tangent to $y = 8x^3 - 22x^2 + 13x - 2$ at $(x, y)$ is $24x^2 - 44x + 13$, and that the bend points are $(0.37, 0.203)$, $(1.46, -5.03)$, approximately. Draw the graph.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly", "core.graph" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/10", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 10", "problem_latex": "Prove that if $g$ and~$k$ are polynomials in~$x$, the derivative of $gk$ is $g'k + gk'$. Hint:\nmultiply the members of $g(x+h) = g(x) + g'(x)h + \\dotsb$ and $k(x+h) = k(x) + k'(x)h + \\dotsb$\nand use~\\Eq{8} for $f = gk$.", "markdown": "Prove that if $g$ and $k$ are polynomials in $x$, the derivative of $gk$ is $g'k + gk'$. Hint: multiply the members of $g(x+h) = g(x) + g'(x)h + \\dotsb$ and $k(x+h) = k(x) + k'(x)h + \\dotsb$ and use $(8)$ for $f = gk$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/2", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 2", "problem_latex": "Prove that the bend points of $y = x^3 - 2x - 5$ are $(.82, -6.09)$, $(-.82$, $-3.91)$, % [** PP: Allow line break between coordinates]\napproximately. Draw the graph and locate the real roots.", "markdown": "Prove that the bend points of $y = x^3 - 2x - 5$ are $(.82, -6.09)$, $(-.82$, $-3.91)$, % [** PP: Allow line break between coordinates] approximately. Draw the graph and locate the real roots.", "answer_latex": [ "$2.1$." ], "answer_markdown": [ "$2.1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 2*x - 5, 0)", "answer_expr": "2.1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 2*x - 5, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page99/1" ], "needs": [ "core.graph", "core.solve.num" ], "expectation": "X=2.1", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/3a", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 3, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 3a", "problem_latex": "Find the bend points of $y = x^3 + 6x^2 + 8x + 8$. Locate the real roots.", "markdown": "Find the bend points of $y = x^3 + 6x^2 + 8x + 8$. Locate the real roots.", "answer_latex": [ "$(-0.845, 4.921)$, $(-3.155, 11.079)$" ], "answer_markdown": [ "$(-0.845, 4.921)$, $(-3.155, 11.079)$" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/3b", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 3, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 3b", "problem_latex": "Find the bend points of $y = x^3 + 6x^2 + 8x + 8$. Locate the real roots.", "markdown": "Find the bend points of $y = x^3 + 6x^2 + 8x + 8$. Locate the real roots.", "answer_latex": [ "between $-4$ and~$-5$." ], "answer_markdown": [ "between $-4$ and $-5$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.solve.num", "core.table" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/4", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 4", "problem_latex": "Locate the real roots of $f(x) = x^4 + x^3 - x - 2 = 0$.\n\nHints: The abscissas of the bend points are the roots of $f'(x) = 4x^3 + 3x^2 - 1 = 0$.\nThe bend points of $y = f'(x)$ are $(0, -1)$ and $(-\\frac{1}{2}, -\\frac{3}{4})$, so that $f'(x)= 0$ has a single real\nroot (it is just less than $\\frac{1}{2}$). The single bend point of $y=f(x)$ is $(\\frac{1}{2}, -\\frac{37}{16})$, approximately.", "markdown": "Locate the real roots of $f(x) = x^4 + x^3 - x - 2 = 0$. Hints: The abscissas of the bend points are the roots of $f'(x) = 4x^3 + 3x^2 - 1 = 0$. The bend points of $y = f'(x)$ are $(0, -1)$ and $(-\\frac{1}{2}, -\\frac{3}{4})$, so that $f'(x)= 0$ has a single real root (it is just less than $\\frac{1}{2}$). The single bend point of $y=f(x)$ is $(\\frac{1}{2}, -\\frac{37}{16})$, approximately.", "answer_latex": [ "$1.1$, $-1.3$." ], "answer_markdown": [ "$1.1$, $-1.3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 + x**3 - x - 2, 0)", "answer_expr": "[1.1, -1.3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 + x**3 - x - 2, 0)" ], "shape": [ "solve: Eq(N - x + 2*x**N, 0)" ], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/5", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 5", "problem_latex": "Locate the real roots of $x^6 - 7x^4 - 3x^2 + 7 = 0$.", "markdown": "Locate the real roots of $x^6 - 7x^4 - 3x^2 + 7 = 0$.", "answer_latex": [ "Between $0$ and~$1$, $0$ and~$-1$, $2.5$ and~$3$, $-2.5$ and~$-3$." ], "answer_markdown": [ "Between $0$ and $1$, $0$ and $-1$, $2.5$ and $3$, $-2.5$ and $-3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.solve.num", "core.table" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/6", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 6", "problem_latex": "Prove that $f''(x)$, given by~\\Eq{7}, is equal to the first derivative of~$f'(x)$.", "markdown": "Prove that $f''(x)$, given by $(7)$, is equal to the first derivative of $f'(x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/7", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 7", "problem_latex": "If $f(x) = f_1(x) + f_2(x)$, prove that the $k$th derivative of~$f$ is equal to the sum of\nthe $k$th derivatives of $f_1$ and~$f_2$. Use~\\Eq{8}.", "markdown": "If $f(x) = f_1(x) + f_2(x)$, prove that the $k$th derivative of $f$ is equal to the sum of the $k$th derivatives of $f_1$ and $f_2$. Use $(8)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/8", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 8", "problem_latex": "Prove that $f^{(k)}(x)$ is equal to the first derivative of $f^{(k-1)}(x)$. Hint: prove this\nfor $f = ax^m$; then prove that it is true for $f=f_1 + f_2$ if true for $f_1$ and~$f_2$.", "markdown": "Prove that $f^{(k)}(x)$ is equal to the first derivative of $f^{(k-1)}(x)$. Hint: prove this for $f = ax^m$; then prove that it is true for $f=f_1 + f_2$ if true for $f_1$ and $f_2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/9a", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 9, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 9a", "problem_latex": "Find the third derivative of $x^6 + 5x^4$ by forming successive first derivatives;\nalso that of $2x^5 - 7x^3 + x$.", "markdown": "Find the third derivative of $x^6 + 5x^4$ by forming successive first derivatives; also that of $2x^5 - 7x^3 + x$.", "answer_latex": [ "$120(x^3 + x)$" ], "answer_markdown": [ "$120(x^3 + x)$" ], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISMATCH", "judge_why": [ "669.0617283950617284", "1804.4444444444444444" ], "problem_expr": "x**6 + 5*x**4", "answer_expr": "120*(x**3 + x)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "differentiate: x**6 + 5*x**4" ], "shape": [ "differentiate: N*x**N + x**N" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page59/9b", "set": "dickson-theory-of-equations-1922/ex-page59", "number": 9, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "59", "location": "Exercise Page59, problem 9b", "problem_latex": "Find the third derivative of $x^6 + 5x^4$ by forming successive first derivatives;\nalso that of $2x^5 - 7x^3 + x$.", "markdown": "Find the third derivative of $x^6 + 5x^4$ by forming successive first derivatives; also that of $2x^5 - 7x^3 + x$.", "answer_latex": [ "$120x^2 - 42$." ], "answer_markdown": [ "$120x^2 - 42$." ], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISMATCH", "judge_why": [ "1213.2994816", "1411.248" ], "problem_expr": "2*x**5 - 7*x**3 + x", "answer_expr": "120*x**2 - 42" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "differentiate: 2*x**5 - 7*x**3 + x" ], "shape": [ "differentiate: 2*N*x**N + x" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/1", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 1", "problem_latex": "Verify that $R_2 = \\omega R_1$, $R_3 = \\omega^2 R_1$. Verify that $R_1$ is a cube root of $8 (\\cos 45°+\ni \\sin 45°)$ by cubing~$R_1$ and applying De Moivre's theorem. Why are the new expressions\nfor~$R_2$ and~$R_3$ evidently also cube roots?", "markdown": "Verify that $R_2 = \\omega R_1$, $R_3 = \\omega^2 R_1$. Verify that $R_1$ is a cube root of $8 (\\cos 45°+ i \\sin 45°)$ by cubing $R_1$ and applying De Moivre’s theorem. Why are the new expressions for $R_2$ and $R_3$ evidently also cube roots?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/2a", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 2, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 2a", "problem_latex": "Find the three cube roots of~$-27$; those of~$-i$; those of~$\\omega$.", "markdown": "Find the three cube roots of $-27$; those of $-i$; those of $\\omega$.", "answer_latex": [ "$-3$, $-3\\omega$, $-3\\omega^2$" ], "answer_markdown": [ "$-3$, $-3\\omega$, $-3\\omega^2$" ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**3, -27) fails at {x: -3*exp(2*pi*I/3)}: leaves 27.0 - 27.0*exp(2*pi*I)", "problem_expr": "Eq(x**3, -27)", "answer_expr": "[-3, -3*exp(2*pi*I/3), -3*exp(4*pi*I/3)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**3, -27)" ], "shape": [ "solve: Eq(x**N, N)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/2b", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 2, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 2b", "problem_latex": "Find the three cube roots of~$-27$; those of~$-i$; those of~$\\omega$.", "markdown": "Find the three cube roots of $-27$; those of $-i$; those of $\\omega$.", "answer_latex": [ "$i$, $\\omega i$;\\quad $\\omega^2 i$" ], "answer_markdown": [ "$i$, $\\omega i$; $\\omega^2 i$" ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**3, -I) fails at {x: I}: ('17.131768489017009727', '0.0')", "problem_expr": "Eq(x**3, -I)", "answer_expr": "[I, exp(2*pi*I/3)*I, exp(4*pi*I/3)*I]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**3, -a)" ], "shape": [ "solve: Eq(x**N, -a)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/2c", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 2, "part": "c", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 2c", "problem_latex": "Find the three cube roots of~$-27$; those of~$-i$; those of~$\\omega$.", "markdown": "Find the three cube roots of $-27$; those of $-i$; those of $\\omega$.", "answer_latex": [ "$R = \\cos 40° + i\\sin 40°$, $\\omega R$, $\\omega^2 R$" ], "answer_markdown": [ "$R = \\cos 40° + i\\sin 40°$, $\\omega R$, $\\omega^2 R$" ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**3, exp(2*pi*I/3)) fails at {x: exp(8*pi*I/9)}: ('368.79291166889901387', '0.0')", "problem_expr": "Eq(x**3, exp(2*pi*I/3))", "answer_expr": "[exp(2*pi*I/9), exp(2*pi*I/9)*exp(2*pi*I/3), exp(2*pi*I/9)*exp(4*pi*I/3)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**3, exp(2*pi*a/3))" ], "shape": [ "solve: Eq(x**N, exp(pi*N*a))" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/3a", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 3, "part": "a", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 3a", "problem_latex": "Find the two square roots of~$i$; those of~$-i$; those of~$\\omega$.", "markdown": "Find the two square roots of $i$; those of $-i$; those of $\\omega$.", "answer_latex": [ "$±(1 + i)/\\sqrt{2}$" ], "answer_markdown": [ "$±(1 + i)/\\sqrt{2}$" ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**2, I) fails at {x: sqrt(2)*(I + 1)/2}: ('0.96492346938775510204', '0.0')", "problem_expr": "Eq(x**2, I)", "answer_expr": "[(1 + I)/sqrt(2), -(1 + I)/sqrt(2)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**2, a)" ], "shape": [ "solve: Eq(x**N, a)" ], "same_problem_in": [], "needs": [ "cas.simplify", "cas.solve.complex", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/3b", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 3, "part": "b", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 3b", "problem_latex": "Find the two square roots of~$i$; those of~$-i$; those of~$\\omega$.", "markdown": "Find the two square roots of $i$; those of $-i$; those of $\\omega$.", "answer_latex": [ "$±(1 - i)/\\sqrt{2}$" ], "answer_markdown": [ "$±(1 - i)/\\sqrt{2}$" ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**2, -I) fails at {x: sqrt(2)*(1 - I)/2}: ('2.0005951200158698671', '0.0')", "problem_expr": "Eq(x**2, -I)", "answer_expr": "[(1 - I)/sqrt(2), -(1 - I)/sqrt(2)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**2, -a)" ], "shape": [ "solve: Eq(x**N, -a)" ], "same_problem_in": [], "needs": [ "cas.simplify", "cas.solve.complex", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/3c", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 3, "part": "c", "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 3c", "problem_latex": "Find the two square roots of~$i$; those of~$-i$; those of~$\\omega$.", "markdown": "Find the two square roots of $i$; those of $-i$; those of $\\omega$.", "answer_latex": [ "$±\\omega^2$" ], "answer_markdown": [ "$±\\omega^2$" ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**2, exp(2*pi*I/3)) fails at {x: exp(4*pi*I/3)}: ('9647.1534004425690253', '0.0')", "problem_expr": "Eq(x**2, exp(2*pi*I/3))", "answer_expr": "[exp(4*pi*I/3), -exp(4*pi*I/3)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**2, exp(2*pi*a/3))" ], "shape": [ "solve: Eq(x**N, exp(pi*N*a))" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/4", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 4", "problem_latex": "Prove that the numbers $\\cos\\theta + i \\sin\\theta$ and no others are represented by points\non the circle of radius unity whose center is the origin.", "markdown": "Prove that the numbers $\\cos\\theta + i \\sin\\theta$ and no others are represented by points on the circle of radius unity whose center is the origin.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/5", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 5", "problem_latex": "If $a+bi$ and $c+di$ are represented by the points~$A$ and~$C$ in Fig.~3, prove that\ntheir sum is represented by the fourth vertex~$S$ of the parallelogram two of whose sides\nare~$OA$ and~$OC$. Hence show that the modulus of the sum of two complex numbers\nis equal to or less than the sum of their moduli, and is equal to or greater than the difference\nof their moduli.", "markdown": "If $a+bi$ and $c+di$ are represented by the points $A$ and $C$ in Fig. 3, prove that their sum is represented by the fourth vertex $S$ of the parallelogram two of whose sides are $OA$ and $OC$. Hence show that the modulus of the sum of two complex numbers is equal to or less than the sum of their moduli, and is equal to or greater than the difference of their moduli.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/6", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 6", "problem_latex": "Let $r$ and~$r'$ be the moduli and $\\theta$ and~$\\alpha$ the amplitudes of two complex numbers\nrepresented by the points $A$ and~$C$ in Fig.~4. Let~$U$ be the point on the $x$-axis one\nunit to the right of the origin~$O$. Construct triangle $OCP$ similar to triangle $OUA$ and\nsimilarly placed, so that corresponding sides are $OC$ and~$OU, CP$ and~$UA$, $OP$ and~$OA$,\nwhile the vertices $O$, $C$, $P$ are in the same order (clockwise or counter-clockwise) as\nthe corresponding vertices $O$, $U$, $A$. Prove that~$P$ represents the product~(§5) of the\ncomplex numbers represented by $A$ and~$C$.", "markdown": "Let $r$ and $r'$ be the moduli and $\\theta$ and $\\alpha$ the amplitudes of two complex numbers represented by the points $A$ and $C$ in Fig. 4. Let $U$ be the point on the $x$-axis one unit to the right of the origin $O$. Construct triangle $OCP$ similar to triangle $OUA$ and similarly placed, so that corresponding sides are $OC$ and $OU, CP$ and $UA$, $OP$ and $OA$, while the vertices $O$, $C$, $P$ are in the same order (clockwise or counter-clockwise) as the corresponding vertices $O$, $U$, $A$. Prove that $P$ represents the product (§5) of the complex numbers represented by $A$ and $C$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/7", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 7", "problem_latex": "If $a+bi$ and $e+fi$ are represented by the points $A$ and~$S$ in Fig.~3, prove that\nthe complex number obtained by subtracting $a+bi$ from $e+fi$ is represented by the point~$C$.\nHence show that the absolute value of the difference of two complex numbers is\nequal to or less than the sum of their absolute values, and is equal to or greater than\nthe difference of their absolute values.", "markdown": "If $a+bi$ and $e+fi$ are represented by the points $A$ and $S$ in Fig. 3, prove that the complex number obtained by subtracting $a+bi$ from $e+fi$ is represented by the point $C$. Hence show that the absolute value of the difference of two complex numbers is equal to or less than the sum of their absolute values, and is equal to or greater than the difference of their absolute values.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page6/8", "set": "dickson-theory-of-equations-1922/ex-page6", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "6", "location": "Exercise Page6, problem 8", "problem_latex": "By modifying Ex.~6, show how to construct geometrically the quotient of two\ncomplex numbers.", "markdown": "By modifying Ex. 6, show how to construct geometrically the quotient of two complex numbers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page62/1", "set": "dickson-theory-of-equations-1922/ex-page62", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "Exercise Page62, problem 1", "problem_latex": "Prove that $x^3 - 7x^2 + 15x - 9 = 0$ has a double root.", "markdown": "Prove that $x^3 - 7x^2 + 15x - 9 = 0$ has a double root.", "answer_latex": [ "$3$." ], "answer_markdown": [ "$3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**3 - 7*x**2 + 15*x - 9", "answer_expr": "3" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pgcd", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page62/2", "set": "dickson-theory-of-equations-1922/ex-page62", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "Exercise Page62, problem 2", "problem_latex": "Show that $x^4 - 8x^2 + 16 = 0$ has two double roots.", "markdown": "Show that $x^4 - 8x^2 + 16 = 0$ has two double roots.", "answer_latex": [ "$2$, $-2$." ], "answer_markdown": [ "$2$, $-2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 - 8*x**2 + 16", "answer_expr": "[2, -2]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page62/3", "set": "dickson-theory-of-equations-1922/ex-page62", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "Exercise Page62, problem 3", "problem_latex": "Prove that $x^4 - 6x^2 - 8x - 3 = 0$ has a triple root.", "markdown": "Prove that $x^4 - 6x^2 - 8x - 3 = 0$ has a triple root.", "answer_latex": [ "$-1$." ], "answer_markdown": [ "$-1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 - 6*x**2 - 8*x - 3", "answer_expr": "-1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page62/4", "set": "dickson-theory-of-equations-1922/ex-page62", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "Exercise Page62, problem 4", "problem_latex": "Test $x^4 - 8x^3 + 22x^2 - 24x + 9 = 0$ for multiple roots.", "markdown": "Test $x^4 - 8x^3 + 22x^2 - 24x + 9 = 0$ for multiple roots.", "answer_latex": [ "Double roots, $1$, $3$." ], "answer_markdown": [ "Double roots, $1$, $3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 - 8*x**3 + 22*x**2 - 24*x + 9", "answer_expr": "[1, 3]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page62/5", "set": "dickson-theory-of-equations-1922/ex-page62", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "Exercise Page62, problem 5", "problem_latex": "Test $x^3 - 6x^2 + 11x - 6 = 0$ for multiple roots.", "markdown": "Test $x^3 - 6x^2 + 11x - 6 = 0$ for multiple roots.", "answer_latex": [ "None." ], "answer_markdown": [ "None." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**3 - 6*x**2 + 11*x - 6", "answer_expr": "[]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page62/6", "set": "dickson-theory-of-equations-1922/ex-page62", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "62", "location": "Exercise Page62, problem 6", "problem_latex": "Test $x^4 - 9x^3 + 9x^2 + 81x - 162 = 0$ for multiple roots.", "markdown": "Test $x^4 - 9x^3 + 9x^2 + 81x - 162 = 0$ for multiple roots.", "answer_latex": [ "$3$, $3$, $-3$, $6$." ], "answer_markdown": [ "$3$, $3$, $-3$, $6$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 - 9*x**3 + 9*x**2 + 81*x - 162", "answer_expr": "[3, 3, -3, 6]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.factor", "cas.pgcd", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page64/1", "set": "dickson-theory-of-equations-1922/ex-page64", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "64", "location": "Exercise Page64, problem 1", "problem_latex": "If $f(x) = 3x^5 + 5x^3 + 4$, the only real root of $f'(x)=0$ is $x = 0$. Show that $(0, 4)$\ninflexion point, and thus that there is no bend point and hence that $f(x)=0$ has a\nsingle real root.", "markdown": "If $f(x) = 3x^5 + 5x^3 + 4$, the only real root of $f'(x)=0$ is $x = 0$. Show that $(0, 4)$ inflexion point, and thus that there is no bend point and hence that $f(x)=0$ has a single real root.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page64/2", "set": "dickson-theory-of-equations-1922/ex-page64", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "64", "location": "Exercise Page64, problem 2", "problem_latex": "Prove that $x^3 - 3x^2 + 3x + c = 0$ has an inflexion point, but no bend point.", "markdown": "Prove that $x^3 - 3x^2 + 3x + c = 0$ has an inflexion point, but no bend point.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page64/3", "set": "dickson-theory-of-equations-1922/ex-page64", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "64", "location": "Exercise Page64, problem 3", "problem_latex": "Show that $x^5 - 10x^3 - 20x^2 - 15x + c = 0$ has two bend points and no horizontal\ninflexion tangents.", "markdown": "Show that $x^5 - 10x^3 - 20x^2 - 15x + c = 0$ has two bend points and no horizontal inflexion tangents.", "answer_latex": [ "Use Ex.~3, p.~62, abscissas $-1$, $3$." ], "answer_markdown": [ "Use Ex. 3, p. 62, abscissas $-1$, $3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page64/4", "set": "dickson-theory-of-equations-1922/ex-page64", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "64", "location": "Exercise Page64, problem 4", "problem_latex": "Prove that $3x^5 - 40x^3 + 240x + c = 0$ has no bend point, but has two horizontal\ninflexion tangents.", "markdown": "Prove that $3x^5 - 40x^3 + 240x + c = 0$ has no bend point, but has two horizontal inflexion tangents.", "answer_latex": [ "Use Ex.~2, p.~62." ], "answer_markdown": [ "Use Ex. 2, p. 62." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page64/5", "set": "dickson-theory-of-equations-1922/ex-page64", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "64", "location": "Exercise Page64, problem 5", "problem_latex": "Prove that any function $x^3 - 3\\alpha x^2 + \\dotsb$ of the third degree can be written in\n\\index{Cubic equation!reduced}% [** PP: Not using page range]\nthe form $f(x) = (x-\\alpha)^3 + ax + b$. The straight line having the equation $y = ax+b$ meets\nthe graph of $y=f(x)$ in three coincident points with the abscissa $\\alpha$ and hence is an\ninflexion tangent. If we take new axes of coordinates parallel to the old and intersecting\nat the new origin $(\\alpha, 0)$, i.e., if we make the transformation $x = X+\\alpha$, $y = Y$,\n%% -----File: 071.png---Folio 65-------\nof coordinates, we see that the equation $f(x)=0$ becomes a reduced cubic equation\n$X^3 + pX + q = 0$~(§42).", "markdown": "Prove that any function $x^3 - 3\\alpha x^2 + \\dotsb$ of the third degree can be written in % [** PP: Not using page range] the form $f(x) = (x-\\alpha)^3 + ax + b$. The straight line having the equation $y = ax+b$ meets the graph of $y=f(x)$ in three coincident points with the abscissa $\\alpha$ and hence is an inflexion tangent. If we take new axes of coordinates parallel to the old and intersecting at the new origin $(\\alpha, 0)$, i.e., if we make the transformation $x = X+\\alpha$, $y = Y$, %% -----File: 071.png---Folio 65------- of coordinates, we see that the equation $f(x)=0$ becomes a reduced cubic equation $X^3 + pX + q = 0$ (§42).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page64/6", "set": "dickson-theory-of-equations-1922/ex-page64", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "64", "location": "Exercise Page64, problem 6", "problem_latex": "Find the inflexion tangent to $y = x^3 + 6x^2 - 3x + 1$ and transform\n$x^3 + 6x^2 - 3x + 1 = 0$ into a reduced cubic equation.", "markdown": "Find the inflexion tangent to $y = x^3 + 6x^2 - 3x + 1$ and transform $x^3 + 6x^2 - 3x + 1 = 0$ into a reduced cubic equation.", "answer_latex": [ "$y = -15x - 7$, $X^3 - 15X + 23 = 0$." ], "answer_markdown": [ "$y = -15x - 7$, $X^3 - 15X + 23 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.expand", "cas.solve.poly", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/1", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 1", "problem_latex": "$x^3 + 2x - 4 = 0$.", "markdown": "$x^3 + 2x - 4 = 0$.", "answer_latex": [ "One real." ], "answer_markdown": [ "One real." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.graph" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/10", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 10", "problem_latex": "Prove that no straight line crosses the graph of $y = f(x)$ in more than $n$~points if\nthe degree~$n$ of the real polynomial $f(x)$ exceeds unity. [Apply~§16.] This fact serves as a check on the accuracy of a graph.", "markdown": "Prove that no straight line crosses the graph of $y = f(x)$ in more than $n$ points if the degree $n$ of the real polynomial $f(x)$ exceeds unity. [Apply §16.] This fact serves as a check on the accuracy of a graph.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/2", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 2", "problem_latex": "$x^3 - 7x + 7 = 0$.", "markdown": "$x^3 - 7x + 7 = 0$.", "answer_latex": [ "$(±\\sqrt{\\frac{7}{3}}, 7\\mp\\frac{14}{3}\\sqrt{\\frac{7}{3}})$, three real." ], "answer_markdown": [ "$(±\\sqrt{\\frac{7}{3}}, 7\\mp\\frac{14}{3}\\sqrt{\\frac{7}{3}})$, three real." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.subst", "core.graph" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/3", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 3", "problem_latex": "$x^3 - 2x - 1 = 0$.", "markdown": "$x^3 - 2x - 1 = 0$.", "answer_latex": [ "$(±\\sqrt\\frac{2}{3}, -1\\mp\\frac{4}{3}\\sqrt{\\frac{2}{3}})$, three." ], "answer_markdown": [ "$(±\\sqrt\\frac{2}{3}, -1\\mp\\frac{4}{3}\\sqrt{\\frac{2}{3}})$, three." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.subst", "core.graph" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/4", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 4", "problem_latex": "$x^3 + 6x^2 - 3x + 1 = 0$.", "markdown": "$x^3 + 6x^2 - 3x + 1 = 0$.", "answer_latex": [ "$(-2±\\sqrt{5}, 23\\mp10\\sqrt{5})$, one." ], "answer_markdown": [ "$(-2±\\sqrt{5}, 23\\mp10\\sqrt{5})$, one." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.subst", "core.graph" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/5", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 5", "problem_latex": "Prove that the inflexion point of $y = x^3 - 3lx + q$ is $(0, q)$.", "markdown": "Prove that the inflexion point of $y = x^3 - 3lx + q$ is $(0, q)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/6", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 6", "problem_latex": "Show that the theorem in the text is equivalent to that in~§45.", "markdown": "Show that the theorem in the text is equivalent to that in §45.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/7", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 7", "problem_latex": "Prove that, if $m$ and~$n$ are positive odd integers and $m>n$, $x^m + px^n + q = 0$ has\nno bend point and hence has a single real root if $p>0$; but, if $p<0$, it has just two\nbend points which are on the same side or opposite sides of the $x$-axis according as\n\\[\n\\left(\\frac{np}{m}\\right)^m + \\left(\\frac{nq}{m-n}\\right)^{m-n}\n\\]\nis positive or negative, so that the number of real roots is $1$ or~$3$ in the respective cases.", "markdown": "Prove that, if $m$ and $n$ are positive odd integers and $m>n$, $x^m + px^n + q = 0$ has no bend point and hence has a single real root if $p>0$; but, if $p<0$, it has just two bend points which are on the same side or opposite sides of the $x$-axis according as (npm)^m + (nqm-n)^m-n is positive or negative, so that the number of real roots is $1$ or $3$ in the respective cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/8", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 8", "problem_latex": "Draw the graph of $y = x^4 - x^2$. By finding its intersections with the line $y = mx + b$, solve $x^4 - x^2 - mx - b= 0$.", "markdown": "Draw the graph of $y = x^4 - x^2$. By finding its intersections with the line $y = mx + b$, solve $x^4 - x^2 - mx - b= 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**4 - x**2 - m*x - b, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(-a - b*x + x**4 - x**2, 0)" ], "shape": [ "solve: Eq(-a - b*x, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.graph" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page66/9", "set": "dickson-theory-of-equations-1922/ex-page66", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "66", "location": "Exercise Page66, problem 9", "problem_latex": "Prove that, if $p$ and~$q$ are positive, $x^{2m} - px^{2n} + q = 0$ has four distinct real roots,\ntwo pairs of equal roots, or no real root, according as\n\\[\n\\left(\\frac{np}{m}\\right)^m - \\left(\\frac{nq}{m-n}\\right)^{m-n} > 0,\n\\quad\\text{${} = 0$,\\quad or\\quad ${} < 0$}.\n\\]", "markdown": "Prove that, if $p$ and $q$ are positive, $x^{2m} - px^{2n} + q = 0$ has four distinct real roots, two pairs of equal roots, or no real root, according as (npm)^m - (nqm-n)^m-n > 0, ${} = 0$, or ${} < 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/1", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 1", "problem_latex": "An equation all of whose coefficients are of like sign has no positive root. Why\nis this self-evident?", "markdown": "An equation all of whose coefficients are of like sign has no positive root. Why is this self-evident?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/10", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 10", "problem_latex": "Prove that we obtain an upper limit to the number of real roots of $f(x)=0$\nbetween $a$ and~$b$, if we set\n\\[\nx = \\frac{a+by}{1+y}\\qquad\n\\left(\\therefore y=\\frac{x-a}{b-x}\\right)\n\\]\nmultiply by $(1+y)^n$, and apply Descartes' rule to the resulting equation in~$y$.", "markdown": "Prove that we obtain an upper limit to the number of real roots of $f(x)=0$ between $a$ and $b$, if we set x = a+by1+y (∴y=x-ab-x) multiply by $(1+y)^n$, and apply Descartes’ rule to the resulting equation in $y$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/11", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 11", "problem_latex": "Show by the method of Ex.~10 that there is a single root between $2$ and~$4$ of\n$x^3 + x^2 - 17x + 15 = 0$. Here we have $27y^3 + 3y^2 - 23y - 7 = 0$.", "markdown": "Show by the method of Ex. 10 that there is a single root between $2$ and $4$ of $x^3 + x^2 - 17x + 15 = 0$. Here we have $27y^3 + 3y^2 - 23y - 7 = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/12", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 12", "problem_latex": "In the astronomical problem of three bodies occurs the equation\n\\[\nr^5 + (3 - \\mu)r^4 + (3 - 2\\mu )r^3 - \\mu r^2 - 2\\mu r - \\mu = 0,\n\\]\nwhere $0 < \\mu < 1$. Why is there a single positive real root?", "markdown": "In the astronomical problem of three bodies occurs the equation r^5 + (3 - )r^4 + (3 - 2)r^3 - r^2 - 2r - = 0, where $0 < \\mu < 1$. Why is there a single positive real root?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/13", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 13, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 13", "problem_latex": "Prove that $x^5 + x^3 - x^2 + 2x - 3 = 0$ has four imaginary roots by applying Descartes'\nrule to the equation in~$y$ whose roots are the squares of the roots of the former.\nTranspose the odd powers, square each new member, and replace $x^2$ by~$y$.", "markdown": "Prove that $x^5 + x^3 - x^2 + 2x - 3 = 0$ has four imaginary roots by applying Descartes’ rule to the equation in $y$ whose roots are the squares of the roots of the former. Transpose the odd powers, square each new member, and replace $x^2$ by $y$.", "answer_latex": [ "$y^5 + 2y^4 + 5y^3 + 3y^2 - 2y - 9 = 0$." ], "answer_markdown": [ "$y^5 + 2y^4 + 5y^3 + 3y^2 - 2y - 9 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/14", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 14", "problem_latex": "As in Ex.~13 prove that $x^3 + x^2 + 8x + 6 = 0$ has imaginary roots.", "markdown": "As in Ex. 13 prove that $x^3 + x^2 + 8x + 6 = 0$ has imaginary roots.", "answer_latex": [ "$y^3 + 15y^2 + 52y - 36 = 0$." ], "answer_markdown": [ "$y^3 + 15y^2 + 52y - 36 = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/15", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 15, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 15", "problem_latex": "If a real equation $f(x)=0$ of degree~$n$ has $n$~real roots, the number of positive\nroots is exactly equal to the number~$V$ of variations of sign. Hint: consider also\n$f(-x)$.", "markdown": "If a real equation $f(x)=0$ of degree $n$ has $n$ real roots, the number of positive roots is exactly equal to the number $V$ of variations of sign. Hint: consider also $f(-x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/16", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 16, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 16", "problem_latex": "Show that $x^3 - x^2 + 2x + 1 = 0$ has no positive root. Hint: multiply by $x + 1$.", "markdown": "Show that $x^3 - x^2 + 2x + 1 = 0$ has no positive root. Hint: multiply by $x + 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/2", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 2", "problem_latex": "There is no negative root of an equation, like $x^5 - 2x^4 - 3x^2 + 7x - 5 = 0$, in which\nthe coefficients of the odd powers of~$x$ are of like sign, and the coefficients of the even\npowers (including the constant term) are of the opposite sign. Verify by taking $x= -p$,\nwhere $p$ is positive.", "markdown": "There is no negative root of an equation, like $x^5 - 2x^4 - 3x^2 + 7x - 5 = 0$, in which the coefficients of the odd powers of $x$ are of like sign, and the coefficients of the even powers (including the constant term) are of the opposite sign. Verify by taking $x= -p$, where $p$ is positive.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/3", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 3", "problem_latex": "$x^3 + a^2 x + b^2 = 0$ has two imaginary roots if $b\\ne 0$.", "markdown": "$x^3 + a^2 x + b^2 = 0$ has two imaginary roots if $b\\ne 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/4", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 4", "problem_latex": "For $n$~even, $x^n - 1 = 0$ has only two real roots.", "markdown": "For $n$ even, $x^n - 1 = 0$ has only two real roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/5", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 5", "problem_latex": "For $n$~odd, $x^n - 1 = 0$ has only one real root.", "markdown": "For $n$ odd, $x^n - 1 = 0$ has only one real root.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/6", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 6", "problem_latex": "For $n$~even, $x^n + 1 = 0$ has no real root; for $n$~odd, only one.", "markdown": "For $n$ even, $x^n + 1 = 0$ has no real root; for $n$ odd, only one.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/7", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 7", "problem_latex": "$x^4 + 12x^2 + 5x - 9 = 0$ has just two imaginary roots.", "markdown": "$x^4 + 12x^2 + 5x - 9 = 0$ has just two imaginary roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/8", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 8", "problem_latex": "$x^4 + a^2 x^2 + b^2 x - c^2 = 0$ ($c\\ne 0$) has just two imaginary roots.", "markdown": "$x^4 + a^2 x^2 + b^2 x - c^2 = 0$ ($c\\ne 0$) has just two imaginary roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page74/9", "set": "dickson-theory-of-equations-1922/ex-page74", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "74", "location": "Exercise Page74, problem 9", "problem_latex": "Descartes' rule enables us to find the exact number of positive roots only when\nall the coefficients are of like sign or when\n\\[\nf(x) = x^n + p_1 x^{n-1} + \\dotsb + p_{n-s} x^s\n - p_{n-s+1} x^{s-1} - \\dotsb - p_n = 0,\n\\]\neach $p_i$ being $\\geqq 0$. Without using that rule, show that the latter equation has one\nand only one positive root~$r$. Hints: There is a positive root~$r$ by~§63 ($a=0$, $b=\\infty$).\nDenote by~$P(x)$ the quotient of the sum of the positive terms by~$x^s$, and by $-N(x)$\nthat of the negative terms. Then $N(x)$ is a sum of powers of~$1/x$ with positive coefficients.\n\\begin{align*}\n\\text{If}\\quad x>r,\\qquad P(x)>P(r),\\qquad N(x)0; \\\\\n\\text{If}\\quad xN(r),\\qquad f(x)<0.\n\\end{align*}", "markdown": "Descartes’ rule enables us to find the exact number of positive roots only when all the coefficients are of like sign or when f(x) = x^n + p_1 x^n-1 + + p_n-s x^s - p_n-s+1 x^s-1 - - p_n = 0, each $p_i$ being $\\geqq 0$. Without using that rule, show that the latter equation has one and only one positive root $r$. Hints: There is a positive root $r$ by §63 ($a=0$, $b=\\infty$). Denote by $P(x)$ the quotient of the sum of the positive terms by $x^s$, and by $-N(x)$ that of the negative terms. Then $N(x)$ is a sum of powers of $1/x$ with positive coefficients. align* If x>r, P(x)>P(r), N(x)0; If xN(r), f(x)<0. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page78/1", "set": "dickson-theory-of-equations-1922/ex-page78", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "78", "location": "Exercise Page78, problem 1", "problem_latex": "$x^3 +2x +20 = 0$.", "markdown": "$x^3 +2x +20 = 0$.", "answer_latex": [ "One, between $-2$ and~$-3$." ], "answer_markdown": [ "One, between $-2$ and $-3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**3 + 2*x + 20", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.table", "other:sturm_sequence" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page78/2", "set": "dickson-theory-of-equations-1922/ex-page78", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "78", "location": "Exercise Page78, problem 2", "problem_latex": "$x^3 +x-3 = 0$.", "markdown": "$x^3 +x-3 = 0$.", "answer_latex": [ "One, between $1$ and~$2$." ], "answer_markdown": [ "One, between $1$ and $2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**3 + x - 3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.table", "other:sturm_sequence" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/1", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 1", "problem_latex": "$x^3 + 3x^2 - 2x - 5 = 0$.", "markdown": "$x^3 + 3x^2 - 2x - 5 = 0$.", "answer_latex": [ "$(-4, -3)$, $(-2, -1)$, $(1, 2)$." ], "answer_markdown": [ "$(-4, -3)$, $(-2, -1)$, $(1, 2)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**3 + 3*x**2 - 2*x - 5", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/2", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 2", "problem_latex": "$x^4 + 12x^2 + 5x - 9 = 0$.", "markdown": "$x^4 + 12x^2 + 5x - 9 = 0$.", "answer_latex": [ "$(-2, -1)$, $(0, 1)$." ], "answer_markdown": [ "$(-2, -1)$, $(0, 1)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 + 12*x**2 + 5*x - 9", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/3", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 3", "problem_latex": "$x^3 - 7x - 7 = 0$.", "markdown": "$x^3 - 7x - 7 = 0$.", "answer_latex": [ "$(-2, -1.5)$, $(-1.5, -1)$, $(3, 4)$." ], "answer_markdown": [ "$(-2, -1.5)$, $(-1.5, -1)$, $(3, 4)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**3 - 7*x - 7", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/4", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 4", "problem_latex": "$3x^4 - 6x^2 + 8x - 3 = 0$.", "markdown": "$3x^4 - 6x^2 + 8x - 3 = 0$.", "answer_latex": [ "$(-2, -1)$, $(0, 1)$." ], "answer_markdown": [ "$(-2, -1)$, $(0, 1)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "3*x**4 - 6*x**2 + 8*x - 3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/5", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 5", "problem_latex": "$x^6 + 6x^5 - 30x^2 - 12x - 9 = 0$ [stop with~$f_2$].", "markdown": "$x^6 + 6x^5 - 30x^2 - 12x - 9 = 0$ [stop with $f_2$].", "answer_latex": [ "$(-7, -6)$, $(1, 2)$." ], "answer_markdown": [ "$(-7, -6)$, $(1, 2)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**6 + 6*x**5 - 30*x**2 - 12*x - 9", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/6", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 6", "problem_latex": "$x^4 - 8x^3 + 25x^2 - 36x + 8 = 0$.", "markdown": "$x^4 - 8x^3 + 25x^2 - 36x + 8 = 0$.", "answer_latex": [ "$(0, 1)$, $(3, 4)$." ], "answer_markdown": [ "$(0, 1)$, $(3, 4)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 - 8*x**3 + 25*x**2 - 36*x + 8", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/7", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 7", "problem_latex": "For $f = x^3 + px + q$ ($p\\ne 0$), show that $f_1 = 3x^2 + p$, $f_2 = -2px - 3q$,\n\\[\n4p^2 f_1 = (-6px + 9q)f_2 - f_3,\\quad\nf_3 = -4p^3 - 27q^2,\n\\]\nso that $f_3$ is the discriminant~$\\Delta$~(§44). Let $[p]$ denote the sign of~$p$. Then the signs\nof $f$, $f_1$, $f_2$, $f_3$ are\n\\begin{align*}\n&{}-{}+{}+ [p]\\ [\\Delta]\\quad \\text{for $x = -\\infty$}, \\\\\n&{}+{}+{}- [p]\\ [\\Delta]\\quad \\text{for $x = +\\infty$}.\n\\end{align*}\nFor $\\Delta$ negative there is a single real root. For $\\Delta$ positive and therefore $p$~negative,\nthere are three distinct real roots. For $\\Delta = 0$, $f_2$~is a divisor of~$f_1$ and~$f$, so that\n$x = -3q/(2p)$ is a double root.", "markdown": "For $f = x^3 + px + q$ ($p\\ne 0$), show that $f_1 = 3x^2 + p$, $f_2 = -2px - 3q$, 4p^2 f_1 = (-6px + 9q)f_2 - f_3, f_3 = -4p^3 - 27q^2, so that $f_3$ is the discriminant $\\Delta$ (§44). Let $[p]$ denote the sign of $p$. Then the signs of $f$, $f_1$, $f_2$, $f_3$ are align* &-++ [p] [] for $x = -\\infty$, &++- [p] [] for $x = +\\infty$. align* For $\\Delta$ negative there is a single real root. For $\\Delta$ positive and therefore $p$ negative, there are three distinct real roots. For $\\Delta = 0$, $f_2$ is a divisor of $f_1$ and $f$, so that $x = -3q/(2p)$ is a double root.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/8", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 8", "problem_latex": "Prove that if one of Sturm's functions has $p$~imaginary roots, the initial equation\nhas at least $p$~imaginary roots.", "markdown": "Prove that if one of Sturm’s functions has $p$ imaginary roots, the initial equation has at least $p$ imaginary roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page79/9", "set": "dickson-theory-of-equations-1922/ex-page79", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "79", "location": "Exercise Page79, problem 9", "problem_latex": "State Sturm's theorem so as to include the possibility of~$a$, or~$b$, or both $a$ and~$b$ being roots of $f(x)=0$.", "markdown": "State Sturm’s theorem so as to include the possibility of $a$, or $b$, or both $a$ and $b$ being roots of $f(x)=0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page83/1", "set": "dickson-theory-of-equations-1922/ex-page83", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "83", "location": "Exercise Page83, problem 1", "problem_latex": "For $f = x^4 - 8x^2 + 16$, prove that $F_1 = x^3 - 4x$, $F_2 = x^2 - 4$, $F_1 = xF_2$. Hence $n = 2$.\nVerify that $V_{-\\infty} = 2$, $V_{\\infty} = 0$, and that there are just two real roots, each a double\nroot.", "markdown": "For $f = x^4 - 8x^2 + 16$, prove that $F_1 = x^3 - 4x$, $F_2 = x^2 - 4$, $F_1 = xF_2$. Hence $n = 2$. Verify that $V_{-\\infty} = 2$, $V_{\\infty} = 0$, and that there are just two real roots, each a double root.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.expand", "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page83/2", "set": "dickson-theory-of-equations-1922/ex-page83", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "83", "location": "Exercise Page83, problem 2", "problem_latex": "$x^4 - 5x^3 + 9x^2 - 7x + 2 = 0$.", "markdown": "$x^4 - 5x^3 + 9x^2 - 7x + 2 = 0$.", "answer_latex": [ "$1$, $1$, $1$, $2$." ], "answer_markdown": [ "$1$, $1$, $1$, $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 5*x**3 + 9*x**2 - 7*x + 2, 0)", "answer_expr": "[1, 1, 1, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 5*x**3 + 9*x**2 - 7*x + 2, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page83/3", "set": "dickson-theory-of-equations-1922/ex-page83", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "83", "location": "Exercise Page83, problem 3", "problem_latex": "$x^4 + 2x^3 - 3x^2 - 4x + 4 = 0$.", "markdown": "$x^4 + 2x^3 - 3x^2 - 4x + 4 = 0$.", "answer_latex": [ "$1$, $1$, $-2$, $-2$." ], "answer_markdown": [ "$1$, $1$, $-2$, $-2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 + 2*x**3 - 3*x**2 - 4*x + 4, 0)", "answer_expr": "[1, 1, -2, -2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 + 2*x**3 - 3*x**2 - 4*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page83/4", "set": "dickson-theory-of-equations-1922/ex-page83", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "83", "location": "Exercise Page83, problem 4", "problem_latex": "$x^4 - x^2 - 2x + 2 = 0$.", "markdown": "$x^4 - x^2 - 2x + 2 = 0$.", "answer_latex": [ "$1$, $1$, two imaginary." ], "answer_markdown": [ "$1$, $1$, two imaginary." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**4 - x**2 - 2*x + 2, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**4 - x**2 - 2*x + 2, 0)" ], "shape": [ "solve: Eq(N*x + N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page85/1", "set": "dickson-theory-of-equations-1922/ex-page85", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Exercise Page85, problem 1", "problem_latex": "$x^3 -x^2 -2x+1=0$.", "markdown": "$x^3 -x^2 -2x+1=0$.", "answer_latex": [ "$(-2, -1)$, $(0, 1)$, $(1, 2)$." ], "answer_markdown": [ "$(-2, -1)$, $(0, 1)$, $(1, 2)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.table" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page85/2", "set": "dickson-theory-of-equations-1922/ex-page85", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Exercise Page85, problem 2", "problem_latex": "$x^3 +3x^2 -2x-5=0$.", "markdown": "$x^3 +3x^2 -2x-5=0$.", "answer_latex": [ "$(-4, -3)$, $(-2, -1)$, $(1, 2)$." ], "answer_markdown": [ "$(-4, -3)$, $(-2, -1)$, $(1, 2)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.table" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page85/3", "set": "dickson-theory-of-equations-1922/ex-page85", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Exercise Page85, problem 3", "problem_latex": "Prove that if $f{(a)}\\ne 0$, $V_a$ equals the number of real roots $>a$ or exceeds that number by an even integer.", "markdown": "Prove that if $f{(a)}\\ne 0$, $V_a$ equals the number of real roots $>a$ or exceeds that number by an even integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page85/4", "set": "dickson-theory-of-equations-1922/ex-page85", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Exercise Page85, problem 4", "problem_latex": "Prove that there is no root greater than a number making each of the functions~\\Eq{12}\npositive, if the leading coefficient of $f(x)$ is positive. (Newton.)", "markdown": "Prove that there is no root greater than a number making each of the functions $(12)$ positive, if the leading coefficient of $f(x)$ is positive. (Newton.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page85/5", "set": "dickson-theory-of-equations-1922/ex-page85", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Exercise Page85, problem 5", "problem_latex": "Hence verify that $x^4 -4x^3 - 3x + 23 = 0$ has no root~$>4$.", "markdown": "Hence verify that $x^4 -4x^3 - 3x + 23 = 0$ has no root $>4$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page85/6", "set": "dickson-theory-of-equations-1922/ex-page85", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "85", "location": "Exercise Page85, problem 6", "problem_latex": "Show that $x^4 - 4x^3 + x^2 + 6x + 2 = 0$ has no root~$>3$.", "markdown": "Show that $x^4 - 4x^3 + x^2 + 6x + 2 = 0$ has no root $>3$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst", "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/1", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 1", "problem_latex": "$x^3 +2x+20=0$.", "markdown": "$x^3 +2x+20=0$.", "answer_latex": [ "Single, $-2.46955$." ], "answer_markdown": [ "Single, $-2.46955$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 2*x + 20, 0)", "answer_expr": "-2.46955" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 2*x + 20, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X=-2.46955", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/10", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 10", "problem_latex": "The real cube root of~$7.976$.", "markdown": "The real cube root of $7.976$.", "answer_latex": [ "$1.997997997$." ], "answer_markdown": [ "$1.997997997$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.99799799666, printed 1.997997997 (half-unit 5.0e-10; correctly rounded at the printed digits: 1.997997997)", "problem_expr": "7.976**(1/3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 997**(1/3)/5" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page96/3" ], "needs": [ "core.arith" ], "expectation": "X#1.997997997,5E-10", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7.976 ENTER 1 ENTER 3 ÷ yˣ" ], "calculator_value": "+1997997996659985299027791689850350E-33", "printed_value": "1.997997997", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page89/11", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 11, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 11", "problem_latex": "The abscissa of the real point of intersection of the conics $y=x^2$, $xy+x+3y-6=0$.", "markdown": "The abscissa of the real point of intersection of the conics $y=x^2$, $xy+x+3y-6=0$.", "answer_latex": [ "$1.094551482$." ], "answer_markdown": [ "$1.094551482$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 3*x**2 + x - 6, 0)", "answer_expr": "{x: 1.094551482}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 3*x**2 + x - 6, 0)" ], "shape": [ "solve: Eq(N*x**N + N + x + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/12", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 12, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 12", "problem_latex": "Find to 3~decimal places the abscissas of the points of intersection of $x^2+y^2=9$,\\n$y=x^2-x$.", "markdown": "Find to 3 decimal places the abscissas of the points of intersection of $x^2+y^2=9$,$y=x^2-x$.", "answer_latex": [ "$2.059$, $-1.228$." ], "answer_markdown": [ "$2.059$, $-1.228$." ], "checks": [ { "task": "solve", "verdict": "FLAG-EXTRACTION", "judge_why": "Eq(x**2 + y**2, 9) fails at {x: 2059/1000}: ('-3.3764359550173010381', '0.0')", "problem_expr": null, "answer_expr": "[2.059, -1.228]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "solve: (Eq(a, x**2 - x), Eq(a**2 + x**2, 9))" ], "shape": [ "solve: (Eq(a, -x + x**N), Eq(a**N + x**N, N))" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/13", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 13, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 13", "problem_latex": "A sphere two feet in diameter is formed of a kind of wood a cubic foot of which\nweighs two-thirds as much as a cubic foot of water (i.e., the specific gravity of the wood\nis~$2/3$). Find to four significant figures the depth~$h$ to which the floating sphere\nwill sink in water.", "markdown": "A sphere two feet in diameter is formed of a kind of wood a cubic foot of which weighs two-thirds as much as a cubic foot of water (i.e., the specific gravity of the wood is $2/3$). Find to four significant figures the depth $h$ to which the floating sphere will sink in water.", "answer_latex": [ "$1.2261$." ], "answer_markdown": [ "$1.2261$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(h**3 - 3*h**2 + Rational(8, 3), 0)", "answer_expr": "1.2261" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x**3 - 3*x**2 + 8/3, 0)" ], "shape": [ "solve: Eq(N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": "X=1.2261", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/14", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 14, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 14", "problem_latex": "If the specific gravity of cork is~$1/4$, find to four significant figures how far a\ncork sphere two feet in diameter will sink in water.", "markdown": "If the specific gravity of cork is $1/4$, find to four significant figures how far a cork sphere two feet in diameter will sink in water.", "answer_latex": [ "$0.6527 = \\text{reciprocal of } 2 \\cos 40°$." ], "answer_markdown": [ "$0.6527 = \\text{reciprocal of } 2 \\cos 40°$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(h**3 - 3*h**2 + 1, 0)", "answer_expr": "0.6527" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x**3 - 3*x**2 + 1, 0)" ], "shape": [ "solve: Eq(N*x**N + x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": "X=0.6527", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/15", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 15, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 15", "problem_latex": "Compute $\\cos 20°$ to four decimal places by use of\n\\[\n\\cos 3A = 4\\cos^3 A - 3\\cos A,\\qquad\n\\cos 60° = \\tfrac{1}{2}.\n\\]", "markdown": "Compute $\\cos 20°$ to four decimal places by use of 3A = 4^3 A - 3A, 60° = 12.", "answer_latex": [ "$0.9397$." ], "answer_markdown": [ "$0.9397$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.939692620786, printed 0.9397 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.9397)", "problem_expr": "cos(pi/9)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: cos(pi/9)" ], "shape": [ "evaluate: cos(pi*N)" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": "X#0.9397,5E-5", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/16", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 16, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 16", "problem_latex": "Three intersecting edges of a rectangular parallelopiped are of lengths $6$,~$8$,\nand $10$~feet. If the volume is increased by $300$~cubic feet by equal elongations of the\nedges, find the elongation to three decimal places.", "markdown": "Three intersecting edges of a rectangular parallelopiped are of lengths $6$, $8$, and $10$ feet. If the volume is increased by $300$ cubic feet by equal elongations of the edges, find the elongation to three decimal places.", "answer_latex": [ "$1.3500$." ], "answer_markdown": [ "$1.3500$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq((6 + x)*(8 + x)*(10 + x), 480 + 300)", "answer_expr": "1.3500" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq((x + 6)*(x + 8)*(x + 10), 780)" ], "shape": [ "solve: Eq((N + x)**3, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num" ], "expectation": "X=1.35", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/17", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 17, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 17", "problem_latex": "Given that the volume of a right circular cylinder is $\\alpha\\pi$ and the total area of\nits surface is~$2\\beta\\pi$, prove that the radius~$r$ of its base is a root of $r^3 - \\beta r + \\alpha = 0$. If $\\alpha = 56$,\n$\\beta = 28$, find to four decimal places the two positive roots~$r$. The corresponding altitude\nis~$\\alpha/r^2$.", "markdown": "Given that the volume of a right circular cylinder is $\\alpha\\pi$ and the total area of its surface is $2\\beta\\pi$, prove that the radius $r$ of its base is a root of $r^3 - \\beta r + \\alpha = 0$. If $\\alpha = 56$, $\\beta = 28$, find to four decimal places the two positive roots $r$. The corresponding altitude is $\\alpha/r^2$.", "answer_latex": [ "$2.7138$, $3.3840$." ], "answer_markdown": [ "$2.7138$, $3.3840$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(r**3 - 28*r + 56, 0)", "answer_expr": "[2.7138, 3.3840]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 28*x + 56, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/18", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 18, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 18", "problem_latex": "What rate of interest is implied in an offer to sell a house for \\$2700 cash, or\nin annual installments each of \\$1000 payable 1,~2, and 3~years from date?", "markdown": "What rate of interest is implied in an offer to sell a house for $2700 cash, or in annual installments each of $1000 payable 1, 2, and 3 years from date?", "answer_latex": [ "$5.46\\%$." ], "answer_markdown": [ "$5.46\\%$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(2700*(1 + r)**3, 1000*(1 + r)**2 + 1000*(1 + r) + 1000)", "answer_expr": "{r: 0.0546}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2700*(x + 1)**3, 1000*x + 1000*(x + 1)**2 + 2000)" ], "shape": [ "solve: Eq(N*(x + 1)**N, N*x + N*(x + 1)**N + N)" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/19", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 19, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 19", "problem_latex": "Find the rate of interest implied in an offer to sell a house for \\$3500 cash, or in annual installments each of \\$1000 payable 1,~2, 3, and 4~years from date.", "markdown": "Find the rate of interest implied in an offer to sell a house for $3500 cash, or in annual installments each of $1000 payable 1, 2, 3, and 4 years from date.", "answer_latex": [ "$5.57\\%$." ], "answer_markdown": [ "$5.57\\%$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(3500*(r + 1)**4, 1000*r + 1000*(r + 1)**3 + 1000*(r + 1)**2 + 2000) fails at {r: 557/100}: leaves 6186901.77903", "problem_expr": "Eq(3500*(1 + r)**4, 1000*((1 + r)**3 + (1 + r)**2 + (1 + r) + 1))", "answer_expr": "{r: 5.57}" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(3500*(x + 1)**4, 1000*x + 1000*(x + 1)**3 + 1000*(x + 1)**2 + 2000)" ], "shape": [ "solve: Eq(N*(x + 1)**N, N*x + 2*N*(x + 1)**N + N)" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/2", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 2", "problem_latex": "$x^3 +3x^2 -2x-5=0$.", "markdown": "$x^3 +3x^2 -2x-5=0$.", "answer_latex": [ "$-1.20164$, $1.33006$, $-3.12842$." ], "answer_markdown": [ "$-1.20164$, $1.33006$, $-3.12842$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 3*x**2 - 2*x - 5, 0)", "answer_expr": "[-1.20164, 1.33006, -3.12842]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 3*x**2 - 2*x - 5, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page49/4" ], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/20", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 20, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 20", "problem_latex": "Find the rate of interest implied in an offer to sell a house for \\$3500 cash, or\n\\$4000 payable in annual installments each of \\$1000, the first payable now.", "markdown": "Find the rate of interest implied in an offer to sell a house for $3500 cash, or $4000 payable in annual installments each of $1000, the first payable now.", "answer_latex": [ "$9.70\\%$." ], "answer_markdown": [ "$9.70\\%$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(2500*(1 + r)**3, 1000*((1 + r)**2 + (1 + r) + 1))", "answer_expr": "{r: 0.0970}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2500*(x + 1)**3, 1000*x + 1000*(x + 1)**2 + 2000)" ], "shape": [ "solve: Eq(N*(x + 1)**N, N*x + N*(x + 1)**N + N)" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/3", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 3", "problem_latex": "$x^3 +x^2 -2x-1=0$.", "markdown": "$x^3 +x^2 -2x-1=0$.", "answer_latex": [ "$1.24698$, $-1.80194$, $-0.44504$." ], "answer_markdown": [ "$1.24698$, $-1.80194$, $-0.44504$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + x**2 - 2*x - 1, 0)", "answer_expr": "[1.24698, -1.80194, -0.44504]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + x**2 - 2*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + 2*x**N - 1, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page49/5" ], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/4", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 4", "problem_latex": "$x^4 +4x^3 -17.5x^2 -18x+58.5=0$.", "markdown": "$x^4 +4x^3 -17.5x^2 -18x+58.5=0$.", "answer_latex": [ "$± 2.1213203$, $\\Neg2.1231056$, $-6.1231056$." ], "answer_markdown": [ "$± 2.1213203$, $\\Neg2.1231056$, $-6.1231056$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 + 4*x**3 - 17.5*x**2 - 18*x + 58.5, 0)", "answer_expr": "[2.1213203, -2.1213203, 2.1231056, -6.1231056]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 + 4*x**3 - 35*x**2/2 - 18*x + 117/2, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/5", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 5", "problem_latex": "$x^4 -11,727x+40,385=0$.", "markdown": "$x^4 -11,727x+40,385=0$.", "answer_latex": [ "$3.45592$, $21.43067$." ], "answer_markdown": [ "$3.45592$, $21.43067$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**4 - 11727*x + 40385, 0)", "answer_expr": "[3.45592, 21.43067]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**4 - 11727*x + 40385, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/6", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 6", "problem_latex": "$x^3 =10$.", "markdown": "$x^3 =10$.", "answer_latex": [ "$2.15443$." ], "answer_markdown": [ "$2.15443$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3, 10)", "answer_expr": "2.15443" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3, 10)" ], "shape": [ "solve: Eq(x**N, N)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X=2.15443", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/7", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 7", "problem_latex": "$x^3 +4x^2 -7=0$.", "markdown": "$x^3 +4x^2 -7=0$.", "answer_latex": [ "$-1.7728656$, $\\Neg1.1642479$, $-3.3913823$." ], "answer_markdown": [ "$-1.7728656$, $\\Neg1.1642479$, $-3.3913823$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 + 4*x**2 - 7, 0)", "answer_expr": "[-1.7728656, 1.1642479, -3.3913823]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 + 4*x**2 - 7, 0)" ], "shape": [ "solve: Eq(N*x**N + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page49/6" ], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/8", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 8", "problem_latex": "$x^3 -7x-7=0$.", "markdown": "$x^3 -7x-7=0$.", "answer_latex": [ "$\\Neg3.0489173$, $-1.3568958$, $-1.6920215$." ], "answer_markdown": [ "$\\Neg3.0489173$, $-1.3568958$, $-1.6920215$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 7*x - 7, 0)", "answer_expr": "[3.0489173, -1.3568958, -1.6920215]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 7*x - 7, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page89/9", "set": "dickson-theory-of-equations-1922/ex-page89", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "89", "location": "Exercise Page89, problem 9", "problem_latex": "The root between $2$ and~$3$ of $x^3 -x-9=0$ (make only 3~transformations).", "markdown": "The root between $2$ and $3$ of $x^3 -x-9=0$ (make only 3 transformations).", "answer_latex": [ "$2.24004099$." ], "answer_markdown": [ "$2.24004099$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - x - 9, 0)", "answer_expr": "2.24004099" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - x - 9, 0)" ], "shape": [ "solve: Eq(N - x + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page96/1" ], "needs": [ "core.solve.num" ], "expectation": "X=2.24004099", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page9/1", "set": "dickson-theory-of-equations-1922/ex-page9", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Exercise Page9, problem 1", "problem_latex": "Simplify the trigonometric forms~\\Eq{6} of the four fourth roots of unity. Check\nthe result by factoring $x^4-1$.", "markdown": "Simplify the trigonometric forms $(6)$ of the four fourth roots of unity. Check the result by factoring $x^4-1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.factor", "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page9/2", "set": "dickson-theory-of-equations-1922/ex-page9", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Exercise Page9, problem 2", "problem_latex": "For $n=6$, show that $R = -\\omega^2$. The sixth roots of unity are the three cube roots\nof unity and their negatives. Check by factoring $x^6-1$.", "markdown": "For $n=6$, show that $R = -\\omega^2$. The sixth roots of unity are the three cube roots of unity and their negatives. Check by factoring $x^6-1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.factor", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page9/3", "set": "dickson-theory-of-equations-1922/ex-page9", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Exercise Page9, problem 3", "problem_latex": "From the point representing $a+bi$, how do you obtain that representing $-(a+bi)$?\nHence derive from Fig.~2 and Ex.~2 the points representing the six sixth roots of unity.\nObtain this result another way.", "markdown": "From the point representing $a+bi$, how do you obtain that representing $-(a+bi)$? Hence derive from Fig. 2 and Ex. 2 the points representing the six sixth roots of unity. Obtain this result another way.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page9/4", "set": "dickson-theory-of-equations-1922/ex-page9", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Exercise Page9, problem 4", "problem_latex": "Find the five fifth roots of~$-1$.", "markdown": "Find the five fifth roots of $-1$.", "answer_latex": [ "$-1$, $\\cos A + i \\sin A$ ($A=36°$, $108°$, $252°$, $324°$)." ], "answer_markdown": [ "$-1$, $\\cos A + i \\sin A$ ($A=36°$, $108°$, $252°$, $324°$)." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x**5, -1) fails at {x: I*sqrt(5/8 - sqrt(5)/8) + 1/4 + sqrt(5)/4}: leaves (0.587785252293*I + 0.809016994375)**5 + 1.0", "problem_expr": "Eq(x**5, -1)", "answer_expr": "[-1, cos(pi/5) + I*sin(pi/5), cos(3*pi/5) + I*sin(3*pi/5), cos(7*pi/5) + I*sin(7*pi/5), cos(9*pi/5) + I*sin(9*pi/5)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**5, -1)" ], "shape": [ "solve: Eq(x**N, -1)" ], "same_problem_in": [], "needs": [ "core.complex", "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page9/5", "set": "dickson-theory-of-equations-1922/ex-page9", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Exercise Page9, problem 5", "problem_latex": "Obtain the trigonometric forms of the nine ninth roots of unity. Which of\nthem are cube roots of unity?", "markdown": "Obtain the trigonometric forms of the nine ninth roots of unity. Which of them are cube roots of unity?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page9/6", "set": "dickson-theory-of-equations-1922/ex-page9", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "9", "location": "Exercise Page9, problem 6", "problem_latex": "Which powers of a ninth root~\\Eq{7} of unity are cube roots of unity?", "markdown": "Which powers of a ninth root $(7)$ of unity are cube roots of unity?", "answer_latex": [ "$R^3$, $R^6$, $R^9$." ], "answer_markdown": [ "$R^3$, $R^6$, $R^9$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page94/1", "set": "dickson-theory-of-equations-1922/ex-page94", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "94", "location": "Exercise Page94, problem 1", "problem_latex": "For $f(x) = x^4 + x^3 - 3x^2 - x - 4$, show by Descartes' rule of signs that $f'(x)=0$\nand $f''(x)=0$ each have a single positive root and that neither has a root between $1$\nand~$2$. Which of the values $1$ and~$2$ should be taken as~$\\beta$?", "markdown": "For $f(x) = x^4 + x^3 - 3x^2 - x - 4$, show by Descartes’ rule of signs that $f'(x)=0$ and $f''(x)=0$ each have a single positive root and that neither has a root between $1$ and $2$. Which of the values $1$ and $2$ should be taken as $\\beta$?", "answer_latex": [ "$2$." ], "answer_markdown": [ "$2$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page94/2", "set": "dickson-theory-of-equations-1922/ex-page94", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "94", "location": "Exercise Page94, problem 2", "problem_latex": "When seeking a root between $2$ and~$3$ of $x^3 - x - 9 = 0$, which value should be\ntaken as~$\\beta$?", "markdown": "When seeking a root between $2$ and $3$ of $x^3 - x - 9 = 0$, which value should be taken as $\\beta$?", "answer_latex": [ "$3$." ], "answer_markdown": [ "$3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page96/1", "set": "dickson-theory-of-equations-1922/ex-page96", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Exercise Page96, problem 1", "problem_latex": "Find to 8~decimal places the root between $2$ and~$3$ of $x^3 - x - 9 = 0$.", "markdown": "Find to 8 decimal places the root between $2$ and $3$ of $x^3 - x - 9 = 0$.", "answer_latex": [ "$2.24004099$." ], "answer_markdown": [ "$2.24004099$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - x - 9, 0)", "answer_expr": "2.24004099" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - x - 9, 0)" ], "shape": [ "solve: Eq(N - x + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page89/9" ], "needs": [ "core.arith", "core.solve.num" ], "expectation": "X=2.24004099", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page96/2", "set": "dickson-theory-of-equations-1922/ex-page96", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Exercise Page96, problem 2", "problem_latex": "Find to 7~decimal places the root between $2$ and~$3$ of $x^3 - 2x^2 - 2 = 0$.", "markdown": "Find to 7 decimal places the root between $2$ and $3$ of $x^3 - 2x^2 - 2 = 0$.", "answer_latex": [ "$2.3593041$." ], "answer_markdown": [ "$2.3593041$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**3 - 2*x**2 - 2, 0)", "answer_expr": "2.3593041" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**3 - 2*x**2 - 2, 0)" ], "shape": [ "solve: Eq(N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num" ], "expectation": "X=2.3593041", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page96/3", "set": "dickson-theory-of-equations-1922/ex-page96", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Exercise Page96, problem 3", "problem_latex": "Find the real cube root of $7.976$ to 6~decimal places.", "markdown": "Find the real cube root of $7.976$ to 6 decimal places.", "answer_latex": [ "$1.997998$." ], "answer_markdown": [ "$1.997998$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.99799799666, printed 1.997998 (half-unit 5.0e-7; correctly rounded at the printed digits: 1.997998)", "problem_expr": "7.976**(1/3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 997**(1/3)/5" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page89/10" ], "needs": [ "core.arith", "core.log" ], "expectation": "X#1.997998,5E-7", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7.976 ENTER 3 GOLD ˣ√y" ], "constants": [ { "value": "3", "source": "the 3 in 'cube root' (cube is index 3, so the x-th root with x = 3)" } ], "calculator_value": "+1997997996659985299027791689850351E-33", "printed_value": "1.997998", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "dickson-theory-of-equations-1922/ex-page96/4", "set": "dickson-theory-of-equations-1922/ex-page96", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Exercise Page96, problem 4", "problem_latex": "Explain by Taylor's expansion of $f(2+d)$ why the values of\n\\[\nf(2),\\qquad f'(2),\\qquad\n\\tfrac{1}{2}f''(2),\\qquad\n\\frac{1}{2·3} f'''(2),\\qquad\n\\frac{1}{2·3·4} f''''(2)\n\\]\nare in reverse order the coefficients of the transformed equation\n\\[\nd^4 + 9d^3 + 27d^2 + 31d + 6 = 0,\n\\]\nobtained in the Example in the text, and printed in heavy type.", "markdown": "Explain by Taylor’s expansion of $f(2+d)$ why the values of f(2), f’(2), 12f”(2), 12·3 f”’(2), 12·3·4 f””(2) are in reverse order the coefficients of the transformed equation d^4 + 9d^3 + 27d^2 + 31d + 6 = 0, obtained in the Example in the text, and printed in heavy type.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.derive", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page96/5", "set": "dickson-theory-of-equations-1922/ex-page96", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "96", "location": "Exercise Page96, problem 5", "problem_latex": "The method commonly used to find the positive square root of~$n$ by a computing\nmachine consists in dividing~$n$ by an assumed approximate value~$a$ of the square root\nand taking half the sum of~$a$ and the quotient as a better approximation. Show that\nthe latter agrees with the value of $a+h$ given by applying Newton's method to\n$f(x) = x^2-n$.", "markdown": "The method commonly used to find the positive square root of $n$ by a computing machine consists in dividing $n$ by an assumed approximate value $a$ of the square root and taking half the sum of $a$ and the quotient as a better approximation. Show that the latter agrees with the value of $a+h$ given by applying Newton’s method to $f(x) = x^2-n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/1", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 1", "problem_latex": "$\\frac{1}{4}$.", "markdown": "$\\frac{1}{4}$.", "answer_latex": [ "$132° 20.7'$." ], "answer_markdown": [ "$132° 20.7'$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "132 + Rational(69, 200)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/10", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 10, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 10", "problem_latex": "Find $x$ to 5~decimal places in $x = 3\\log_e x$.", "markdown": "Find $x$ to 5 decimal places in $x = 3\\log_e x$.", "answer_latex": [ "$1.85718$." ], "answer_markdown": [ "$1.85718$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x, 3*log(x))", "answer_expr": "1.85718" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x, 3*log(x))" ], "shape": [ "solve: Eq(x, N*log(x))" ], "same_problem_in": [], "needs": [ "core.log", "core.solve.num" ], "expectation": "X=1.85718", "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/2", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 2", "problem_latex": "$\\frac{3}{8}$.", "markdown": "$\\frac{3}{8}$.", "answer_latex": [ "$157° 12'$." ], "answer_markdown": [ "$157° 12'$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "157 + Rational(1, 5)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/3", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 3", "problem_latex": "Solve $2x - \\log x = 9$.", "markdown": "Solve $2x - \\log x = 9$.", "answer_latex": [ "$4.8425364$." ], "answer_markdown": [ "$4.8425364$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(2*x - log(x), 9) fails at {x: 12106341/2500000}: leaves -0.892365833049", "problem_expr": "Eq(2*x - log(x), 9)", "answer_expr": "4.8425364" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(2*x - log(x), 9)" ], "shape": [ "solve: Eq(N*x - log(x), N)" ], "same_problem_in": [], "needs": [ "core.log", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/4", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 4", "problem_latex": "Solve $3x - \\log x = 9$.", "markdown": "Solve $3x - \\log x = 9$.", "answer_latex": [ "$3.1668771$." ], "answer_markdown": [ "$3.1668771$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(3*x - log(x), 9) fails at {x: 31668771/10000000}: leaves -0.652114660362", "problem_expr": "Eq(3*x - log(x), 9)", "answer_expr": "3.1668771" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(3*x - log(x), 9)" ], "shape": [ "solve: Eq(N*x - log(x), N)" ], "same_problem_in": [], "needs": [ "core.log", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/5", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 5", "problem_latex": "Find the angle just $>15°$ for which\n$\\frac{1}{2} \\sin x + \\sin 2x = 0.64$.", "markdown": "Find the angle just $>15°$ for which $\\frac{1}{2} \\sin x + \\sin 2x = 0.64$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(sin(x*pi/180)/2 + sin(2*x*pi/180), 0.64)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(sin(pi*x/180)/2 + sin(pi*x/90), 16/25)" ], "shape": [ "solve: Eq(N*sin(pi*N*x) + sin(pi*N*x), N)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page98/7" ], "needs": [ "core.arith", "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/6", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 6, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 6", "problem_latex": "Find the angle just $>72°$ for which\n$x - \\frac{1}{2} \\sin x = \\frac{1}{4} \\pi$.", "markdown": "Find the angle just $>72°$ for which $x - \\frac{1}{2} \\sin x = \\frac{1}{4} \\pi$.", "answer_latex": [ "$72° 17'$." ], "answer_markdown": [ "$72° 17'$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(x - sin(x)/2, pi/4) fails at {x: 4337*pi/10800}: leaves -0.000102503072808", "problem_expr": "Eq(x - sin(x)/2, pi/4)", "answer_expr": "(72 + Rational(17, 60))*pi/180" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x - sin(x)/2, pi/4)" ], "shape": [ "solve: Eq(N*sin(x) + x, pi*N)" ], "same_problem_in": [], "needs": [ "core.const", "core.solve.num", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/7", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 7, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 7", "problem_latex": "Find all solutions of Ex.~5 by replacing $\\sin 2x$ by $2\\sin x\\cos x$, squaring, and\nsolving the quartic equation for~$\\cos x$.", "markdown": "Find all solutions of Ex. 5 by replacing $\\sin 2x$ by $2\\sin x\\cos x$, squaring, and solving the quartic equation for $\\cos x$.", "answer_latex": [ "$15° 16\\tfrac{1}{2}'$,\n $85° 56\\tfrac{1}{2}'$,\n $212° 49'$,\n $225° 57'$." ], "answer_markdown": [ "$15° 16\\tfrac{1}{2}'$, $85° 56\\tfrac{1}{2}'$, $212° 49'$, $225° 57'$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(sin(pi*x/180)/2 + sin(pi*x/90), 16/25) fails at {x: 611/40}: leaves 0.0000161615432723", "problem_expr": "Eq(sin(x*pi/180)/2 + sin(2*x*pi/180), 0.64)", "answer_expr": "[611/40, 10313/120, 12769/60, 13557/60]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(sin(pi*x/180)/2 + sin(pi*x/90), 16/25)" ], "shape": [ "solve: Eq(N*sin(pi*N*x) + sin(pi*N*x), N)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page98/5" ], "needs": [ "cas.expand", "cas.solve.poly", "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/8", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 8, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 8", "problem_latex": "Solve similarly $\\sin x + \\sin 2x = 1.2$.", "markdown": "Solve similarly $\\sin x + \\sin 2x = 1.2$.", "answer_latex": [ "$5° 56\\tfrac{1}{2}'$, $25° 18'$." ], "answer_markdown": [ "$5° 56\\tfrac{1}{2}'$, $25° 18'$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(sin(pi*x/180) + sin(pi*x/90), 6/5) fails at {x: 713/120}: leaves -0.890564582205", "problem_expr": "Eq(sin(x*pi/180) + sin(2*x*pi/180), 1.2)", "answer_expr": "[5 + Rational(113, 120), 25 + Rational(3, 10)]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(sin(pi*x/180) + sin(pi*x/90), 6/5)" ], "shape": [ "solve: Eq(2*sin(pi*N*x), N)" ], "same_problem_in": [], "needs": [ "cas.expand", "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page98/9", "set": "dickson-theory-of-equations-1922/ex-page98", "number": 9, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "98", "location": "Exercise Page98, problem 9", "problem_latex": "Find $x$ to 6~decimal places in $\\sin x = x - 2$.", "markdown": "Find $x$ to 6 decimal places in $\\sin x = x - 2$.", "answer_latex": [ "$2.5541949$." ], "answer_markdown": [ "$2.5541949$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(sin(x), x - 2) fails at {x: 25541949/10000000}: leaves 0.00000192920379925", "problem_expr": "Eq(sin(x), x - 2)", "answer_expr": "2.5541949" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(sin(x), x - 2)" ], "shape": [ "solve: Eq(sin(x), N + x)" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page99/1", "set": "dickson-theory-of-equations-1922/ex-page99", "number": 1, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Exercise Page99, problem 1", "problem_latex": "$z^3 - 2z - 5 = 0$.", "markdown": "$z^3 - 2z - 5 = 0$.", "answer_latex": [ "$-1.04727± 1.13594 i$." ], "answer_markdown": [ "$-1.04727± 1.13594 i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(z**3 - 2*z - 5, 0) fails at {z: 56797*I/50000 - 104727/100000}: leaves -2.27188*I + (1.13594*I - 1.04727)**3 - 2.90546", "problem_expr": "Eq(z**3 - 2*z - 5, 0)", "answer_expr": [ "-1.04727 + 1.13594*I", "-1.04727 - 1.13594*I" ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**3 - 2*x - 5, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page59/2" ], "needs": [ "core.complex", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page99/2", "set": "dickson-theory-of-equations-1922/ex-page99", "number": 2, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Exercise Page99, problem 2", "problem_latex": "$28z^3 + 9z^2 - 1 = 0$.", "markdown": "$28z^3 + 9z^2 - 1 = 0$.", "answer_latex": [ "$-\\frac{2}{7} ± \\frac{1}{7}\\sqrt{3}i$." ], "answer_markdown": [ "$-\\frac{2}{7} ± \\frac{1}{7}\\sqrt{3}i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(28*z**3 + 9*z**2 - 1, 0) fails at {z: sqrt(3)*I/7 - 2/7}: leaves 28.0*(0.247435829653*I - 0.285714285714)**3 + 9.0*(0.247435829653*I - 0.285714285714)**2 - 1.0", "problem_expr": "Eq(28*z**3 + 9*z**2 - 1, 0)", "answer_expr": [ "-2/7 + sqrt(3)*I/7", "-2/7 - sqrt(3)*I/7" ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(28*x**3 + 9*x**2 - 1, 0)" ], "shape": [ "solve: Eq(2*N*x**N - 1, 0)" ], "same_problem_in": [ "dickson-theory-of-equations-1922/ex-page46/4" ], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page99/3", "set": "dickson-theory-of-equations-1922/ex-page99", "number": 3, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Exercise Page99, problem 3", "problem_latex": "$z^4 - 3z^2 - 6z = 2$.", "markdown": "$z^4 - 3z^2 - 6z = 2$.", "answer_latex": [ "$-1±i$." ], "answer_markdown": [ "$-1±i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(z**4 - 3*z**2 - 6*z, 2) fails at {z: I - 1}: leaves -6.0*I + (I - 1.0)**4 - 3.0*(I - 1.0)**2 + 4.0", "problem_expr": "Eq(z**4 - 3*z**2 - 6*z, 2)", "answer_expr": [ "-1 + I", "-1 - I" ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**4 - 3*x**2 - 6*x, 2)" ], "shape": [ "solve: Eq(N*x + N*x**N + x**N, N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page99/4", "set": "dickson-theory-of-equations-1922/ex-page99", "number": 4, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Exercise Page99, problem 4", "problem_latex": "$z^4 - 4z^3 + 11z^2 - 14z + 10 = 0$.", "markdown": "$z^4 - 4z^3 + 11z^2 - 14z + 10 = 0$.", "answer_latex": [ "$1±i$, $1±2i$." ], "answer_markdown": [ "$1±i$, $1±2i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(z**4 - 4*z**3 + 11*z**2 - 14*z + 10, 0) fails at {z: I + 1}: leaves -14.0*I + (I + 1.0)**4 - 4.0*(I + 1.0)**3 + 11.0*(I + 1.0)**2 - 4.0", "problem_expr": "Eq(z**4 - 4*z**3 + 11*z**2 - 14*z + 10, 0)", "answer_expr": [ "1 + I", "1 - I", "1 + 2*I", "1 - 2*I" ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**4 - 4*x**3 + 11*x**2 - 14*x + 10, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "dickson-theory-of-equations-1922/ex-page99/5", "set": "dickson-theory-of-equations-1922/ex-page99", "number": 5, "part": null, "book": "dickson-theory-of-equations-1922", "edition": "John Wiley & Sons, New York (Chapman & Hall, London), copyright 1922 (edition to be confirmed from the copy)", "page": "99", "location": "Exercise Page99, problem 5", "problem_latex": "$z^4 - 4z^3 + 9z^2 - 16z + 20 = 0$. Hint:\n\\[\nE(x) \\equiv x(x - 2)(16x^4 - 64x^3 + 136x^2 - 144x + 65) = 0,\n\\]\nand the last factor becomes $(w^2 + 1)(w^2 + 9)$ for $2x = w + 2$.", "markdown": "$z^4 - 4z^3 + 9z^2 - 16z + 20 = 0$. Hint: E(x) x(x - 2)(16x^4 - 64x^3 + 136x^2 - 144x + 65) = 0, and the last factor becomes $(w^2 + 1)(w^2 + 9)$ for $2x = w + 2$.", "answer_latex": [ "$2±i$, $±2i$." ], "answer_markdown": [ "$2±i$, $±2i$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(z**4 - 4*z**3 + 9*z**2 - 16*z + 20, 0) fails at {z: I + 2}: leaves -16.0*I + (I + 2.0)**4 - 4.0*(I + 2.0)**3 + 9.0*(I + 2.0)**2 - 12.0", "problem_expr": "Eq(z**4 - 4*z**3 + 9*z**2 - 16*z + 20, 0)", "answer_expr": [ "2 + I", "2 - I", "2*I", "-2*I" ] } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(x**4 - 4*x**3 + 9*x**2 - 16*x + 20, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] } ], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }