{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "a93854ef-c338-4788-b6af-9299decd73b6",
   "metadata": {},
   "source": [
    "# 5次式の2次式と3次式の積への因数分解の仕方\n",
    "\n",
    "* 黒木玄"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "84fec713-54b7-41e8-a851-3783380b7c22",
   "metadata": {},
   "outputs": [],
   "source": [
    "using SymPy\n",
    "\n",
    "# Override the Base.show definition of SymPy.jl:\n",
    "# https://github.com/JuliaPy/SymPy.jl/blob/29c5bfd1d10ac53014fa7fef468bc8deccadc2fc/src/types.jl#L87-L105\n",
    "\n",
    "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::SymbolicObject)\n",
    "    print(io, as_markdown(\"\\\\displaystyle \" * sympy.latex(x, mode=\"plain\", fold_short_frac=false)))\n",
    "end\n",
    "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::AbstractArray{Sym})\n",
    "    function toeqnarray(x::Vector{Sym})\n",
    "        a = join([\"\\\\displaystyle \" * sympy.latex(x[i]) for i in 1:length(x)], \"\\\\\\\\\")\n",
    "        \"\"\"\\\\left[ \\\\begin{array}{r}$a\\\\end{array} \\\\right]\"\"\"\n",
    "    end\n",
    "    function toeqnarray(x::AbstractArray{Sym,2})\n",
    "        sz = size(x)\n",
    "        a = join([join(\"\\\\displaystyle \" .* map(sympy.latex, x[i,:]), \"&\") for i in 1:sz[1]], \"\\\\\\\\\")\n",
    "        \"\\\\left[ \\\\begin{array}{\" * repeat(\"r\",sz[2]) * \"}\" * a * \"\\\\end{array}\\\\right]\"\n",
    "    end\n",
    "    print(io, as_markdown(toeqnarray(x)))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "59d10a01-ecd8-4c15-9069-ef402e989736",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "factor_for_all_divisors (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all_coeffs(f, x) = sympy.poly(f, x).all_coeffs()\n",
    "\n",
    "function all_divisors(x)\n",
    "    @assert x != 0\n",
    "    A = typeof(x)[]\n",
    "    for i in 1:abs(x)\n",
    "        if mod(x, i) == 0\n",
    "            push!(A, i, -i)\n",
    "        end\n",
    "    end\n",
    "    A\n",
    "end\n",
    "\n",
    "function factor_for_all_divisors(f, var, constant_term)\n",
    "    A = all_divisors(constant_term)\n",
    "    stack([k, f(var=>k).factor()] for k in A; dims=1)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5ae8f78-2ab1-4393-8769-241b6a887164",
   "metadata": {},
   "source": [
    "## 一般論"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9daa19be-0419-4948-a2fa-c7504c573e46",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(a, b, c, d, e, p, q, r, s, t, u, v, x)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@vars a b c d e p q r s t u v x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ba3bf7e-0a9d-4a45-8b32-d5817b59304f",
   "metadata": {},
   "source": [
    "$t\\ne 0$ と仮定する."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b6d25422-733a-46ef-9193-6581287f1868",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle p x^{4} + q x^{3} + r x^{2} + s x + t + x^{5}$\n"
      ],
      "text/plain": [
       "   4      3      2              5\n",
       "p*x  + q*x  + r*x  + s*x + t + x "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = x^5 + p*x^4 + q*x^3 + r*x^2 + s*x + t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f2bfbd7c-2e65-488b-9e2a-6b35c810553c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a x + b + x^{2}$\n"
      ],
      "text/plain": [
       "           2\n",
       "a*x + b + x "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = x^2 + a*x + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "1624c8e3-14b5-411b-95b3-f0db49e6b10f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle c x^{2} + d x + e + x^{3}$\n"
      ],
      "text/plain": [
       "   2              3\n",
       "c*x  + d*x + e + x "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = x^3 + c*x^2 + d*x + e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da0c7f1c-7775-4eab-97a9-049cd6271269",
   "metadata": {},
   "source": [
    "$f$ を $g$ で割ったときの商と余りを求める."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "886e15ba-302b-400d-a70e-3d01f6c53267",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle \\operatorname{Poly}{\\left( x^{3} + \\left(- a + p\\right) x^{2} + \\left(a^{2} - a p - b + q\\right) x -  a^{3} + a^{2} p + 2 a b - a q - b p + r, x, domain=\\mathbb{Z}\\left[p, q, r, a, b\\right] \\right)}\\\\\\displaystyle \\operatorname{Poly}{\\left( \\left(a^{4} - a^{3} p - 3 a^{2} b + a^{2} q + 2 a b p - a r + b^{2} - b q + s\\right) x + a^{3} b - a^{2} b p - 2 a b^{2} + a b q + b^{2} p - b r + t, x, domain=\\mathbb{Z}\\left[p, q, r, s, t, a, b\\right] \\right)}\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "2-element Vector{Sym}:\n",
       "                                            Poly(x^3 + (-a + p)*x^2 + (a^2 - a*p - b + q)*x - a^3 + a^2*p + 2*a*b - a*q - b*p + r, x, domain='ZZ[p,q,r,a,b]')\n",
       " Poly((a^4 - a^3*p - 3*a^2*b + a^2*q + 2*a*b*p - a*r + b^2 - b*q + s)*x + a^3*b - a^2*b*p - 2*a*b^2 + a*b*q + b^2*p - b*r + t, x, domain='ZZ[p,q,r,s,t,a,b]')"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quotient, remainder = sympy.div(f, g, x)\n",
    "sympy.poly.([quotient, remainder], x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47209929-d47e-44b8-90a5-123fd8afc7bf",
   "metadata": {},
   "source": [
    "$h$ が商に一致することと $c,d,e$ が次のようになることは同値."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6ab7be70-17ce-4a78-a6b5-9f3ac250c4fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle - a + p\\\\\\displaystyle a^{2} - a p - b + q\\\\\\displaystyle - a^{3} + a^{2} p + 2 a b - a q - b p + r\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "4-element Vector{Sym}:\n",
       "                                    1\n",
       "                               -a + p\n",
       "                    a^2 - a*p - b + q\n",
       " -a^3 + a^2*p + 2*a*b - a*q - b*p + r"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_, C, D, E = all_coeffs(quotient, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f7a84b8-d796-4c8d-b651-4498cea23290",
   "metadata": {},
   "source": [
    "$be = t\\ne 0$ ならば $b\\ne 0$ でなければいけない.\n",
    "\n",
    "そのとき, 余りが0になることと以下の2つが0になることは同値である."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "67a90673-0e41-437c-a7ff-b1a1f89f16e3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{3} - a^{2} p - 2 a b + a q + b p - r + \\frac{t}{b}$\n"
      ],
      "text/plain": [
       " 3    2                             t\n",
       "a  - a *p - 2*a*b + a*q + b*p - r + -\n",
       "                                    b"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqrem0 = remainder.coeff(x, 0)/b |> expand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "953d1a3d-84e7-4d15-8986-670a965b469b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{4} - a^{3} p - 3 a^{2} b + a^{2} q + 2 a b p - a r + b^{2} - b q + s$\n"
      ],
      "text/plain": [
       " 4    3        2      2                      2          \n",
       "a  - a *p - 3*a *b + a *q + 2*a*b*p - a*r + b  - b*q + s"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqrem1 = remainder.coeff(x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee63618f-f922-4576-99c7-690c008b7ede",
   "metadata": {},
   "source": [
    "与えられた5次でモニックな整数係数多項式 $f$ の整数係数の2次式 $g$ と3次式 $h$ の積への分解を求めるためには, まず $be=t$ を満たす整数の組 $(b, e)$ の各々について, `eqrem0` を0にする整数 $a$ が存在するかどうかを確認し, 存在する場合に `eqrem1` が0になるかどうかも確認し, `eqrem0` と `eqrem1` の両方を $0$ にするような整数の組 $(a,b)$ を見付ければよい.  (見付からない場合には整数係数の2次式と整数係数の3次式の積に分解不可能なことが証明されたことになる.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef750b26-c6ca-4c90-b491-8bf8c7e687aa",
   "metadata": {},
   "source": [
    "## 雑多な計算メモ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "cac0ecd3-b7ad-4211-ae0c-20e06da0c035",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle a^{2} - a p - b + q + x^{2} + x \\left(- a + p\\right)\\\\\\displaystyle - a^{2} c + a c p + b c - c q + t + x^{2} \\left(- a^{3} + a^{2} p + 2 a b - a q - b p - c + r\\right) + x \\left(- a^{2} b + a b p + a c + b^{2} - b q - c p + s\\right)\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "2-element Vector{Sym}:\n",
       "                                                                                             a^2 - a*p - b + q + x^2 + x*(-a + p)\n",
       " -a^2*c + a*c*p + b*c - c*q + t + x^2*(-a^3 + a^2*p + 2*a*b - a*q - b*p - c + r) + x*(-a^2*b + a*b*p + a*c + b^2 - b*q - c*p + s)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.div(f, x^3 + a*x^2 + b*x + c, x) |> collect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5b48a9bd-4b4c-4ced-b6cf-c56104240f14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle a^{4} - a^{3} p - 3 a^{2} b + a^{2} q + 2 a b p - a r + b^{2} - b q + s + x^{4} + x^{3} \\left(- a + p\\right) + x^{2} \\left(a^{2} - a p - b + q\\right) + x \\left(- a^{3} + a^{2} p + 2 a b - a q - b p + r\\right)\\\\\\displaystyle - a^{4} b + a^{3} b p + 3 a^{2} b^{2} - a^{2} b q - 2 a b^{2} p + a b r - b^{3} + b^{2} q - b s + u + x \\left(- a^{5} + a^{4} p + 4 a^{3} b - a^{3} q - 3 a^{2} b p + a^{2} r - 3 a b^{2} + 2 a b q - a s + b^{2} p - b r + t\\right)\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "2-element Vector{Sym}:\n",
       "                                 a^4 - a^3*p - 3*a^2*b + a^2*q + 2*a*b*p - a*r + b^2 - b*q + s + x^4 + x^3*(-a + p) + x^2*(a^2 - a*p - b + q) + x*(-a^3 + a^2*p + 2*a*b - a*q - b*p + r)\n",
       " -a^4*b + a^3*b*p + 3*a^2*b^2 - a^2*b*q - 2*a*b^2*p + a*b*r - b^3 + b^2*q - b*s + u + x*(-a^5 + a^4*p + 4*a^3*b - a^3*q - 3*a^2*b*p + a^2*r - 3*a*b^2 + 2*a*b*q - a*s + b^2*p - b*r + t)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.div(x^6 + p*x^5 + q*x^4 + r*x^3 + s*x^2 + t*x + u, g, x) |> collect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "8a546d0d-a7f0-4bf6-beae-206e904321a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle - a^{3} + a^{2} p + 2 a b - a q - b p - c + r + x^{3} + x^{2} \\left(- a + p\\right) + x \\left(a^{2} - a p - b + q\\right)\\\\\\displaystyle a^{3} c - a^{2} c p - 2 a b c + a c q + b c p + c^{2} - c r + u + x^{2} \\left(a^{4} - a^{3} p - 3 a^{2} b + a^{2} q + 2 a b p + 2 a c - a r + b^{2} - b q - c p + s\\right) + x \\left(a^{3} b - a^{2} b p - a^{2} c - 2 a b^{2} + a b q + a c p + b^{2} p + 2 b c - b r - c q + t\\right)\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "2-element Vector{Sym}:\n",
       "                                                                                                                                                 -a^3 + a^2*p + 2*a*b - a*q - b*p - c + r + x^3 + x^2*(-a + p) + x*(a^2 - a*p - b + q)\n",
       " a^3*c - a^2*c*p - 2*a*b*c + a*c*q + b*c*p + c^2 - c*r + u + x^2*(a^4 - a^3*p - 3*a^2*b + a^2*q + 2*a*b*p + 2*a*c - a*r + b^2 - b*q - c*p + s) + x*(a^3*b - a^2*b*p - a^2*c - 2*a*b^2 + a*b*q + a*c*p + b^2*p + 2*b*c - b*r - c*q + t)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q63, R63 = sympy.div(x^6 + p*x^5 + q*x^4 + r*x^3 + s*x^2 + t*x + u, x^3 + a*x^2 + b*x + c, x) |> collect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "abbd9173-2a71-4c16-a72a-fd8d83d087f2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle a^{4} - a^{3} p - 3 a^{2} b + a^{2} q + 2 a b p + 2 a c - a r + b^{2} - b q - c p + s\\\\\\displaystyle a^{3} b - a^{2} b p - a^{2} c - 2 a b^{2} + a b q + a c p + b^{2} p + 2 b c - b r - c q + t\\\\\\displaystyle a^{3} c - a^{2} c p - 2 a b c + a c q + b c p + c^{2} - c r + u\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "3-element Vector{Sym}:\n",
       "       a^4 - a^3*p - 3*a^2*b + a^2*q + 2*a*b*p + 2*a*c - a*r + b^2 - b*q - c*p + s\n",
       " a^3*b - a^2*b*p - a^2*c - 2*a*b^2 + a*b*q + a*c*p + b^2*p + 2*b*c - b*r - c*q + t\n",
       "                         a^3*c - a^2*c*p - 2*a*b*c + a*c*q + b*c*p + c^2 - c*r + u"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C63 = all_coeffs(R63, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "8de6ac2a-1a95-41f6-a697-707ab803bd5b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a^{3} c - a^{2} c p + a c q + c^{2} - c r + u}{c \\left(2 a - p\\right)}$\n"
      ],
      "text/plain": [
       " 3      2                2          \n",
       "a *c - a *c*p + a*c*q + c  - c*r + u\n",
       "------------------------------------\n",
       "            c*(2*a - p)             "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B63 = sympy.solve(C63[end], b)[end]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "6290231b-9240-475b-b148-921f92051037",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eqrem63_1 = -(((((C63[1])(b => B63))(u => c * v) * (-2a + p) ^ 2).expand()).factor()) = a^6 - 3*a^5*p + 3*a^4*p^2 + 2*a^4*q - 4*a^3*c - a^3*p^3 - 4*a^3*p*q + 4*a^3*v + 7*a^2*c*p + 2*a^2*p^2*q + a^2*p*r - 5*a^2*p*v + a^2*q^2 - 4*a^2*s - 4*a*c*p^2 - a*p^2*r + 2*a*p^2*v - a*p*q^2 + 4*a*p*s - c^2 + c*p^3 - c*p*q + 2*c*r - 2*c*v - p^2*s + p*q*r - p*q*v - r^2 + 2*r*v - v^2\n"
     ]
    }
   ],
   "source": [
    "@show eqrem63_1 = -(C63[1](b=>B63)(u=>c*v) * (-2*a + p)^2).expand().factor();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "5e0a7857-58b8-4650-a504-d8d112aea9e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle - 3 p\\\\\\displaystyle 3 p^{2} + 2 q\\\\\\displaystyle - 4 c - p^{3} - 4 p q + 4 v\\\\\\displaystyle 7 c p + 2 p^{2} q + p r - 5 p v + q^{2} - 4 s\\\\\\displaystyle - 4 c p^{2} - p^{2} r + 2 p^{2} v - p q^{2} + 4 p s\\\\\\displaystyle - c^{2} + c p^{3} - c p q + 2 c r - 2 c v - p^{2} s + p q r - p q v - r^{2} + 2 r v - v^{2}\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "7-element Vector{Sym}:\n",
       "                                                                                1\n",
       "                                                                             -3*p\n",
       "                                                                      3*p^2 + 2*q\n",
       "                                                         -4*c - p^3 - 4*p*q + 4*v\n",
       "                                        7*c*p + 2*p^2*q + p*r - 5*p*v + q^2 - 4*s\n",
       "                                       -4*c*p^2 - p^2*r + 2*p^2*v - p*q^2 + 4*p*s\n",
       " -c^2 + c*p^3 - c*p*q + 2*c*r - 2*c*v - p^2*s + p*q*r - p*q*v - r^2 + 2*r*v - v^2"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all_coeffs(eqrem63_1, a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "485f8585-85bf-47b4-885f-78b2a9efff7a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{rr}\\displaystyle x^{5} + x + 1&\\displaystyle \\left(x^{2} + x + 1\\right) \\left(x^{3} - x^{2} + 1\\right)\\\\\\displaystyle x^{5} - 5 x + 3&\\displaystyle \\left(x^{2} + x - 1\\right) \\left(x^{3} - x^{2} + 2 x - 3\\right)\\\\\\displaystyle x^{5} + x + 6&\\displaystyle \\left(x^{2} + x + 2\\right) \\left(x^{3} - x^{2} - x + 3\\right)\\\\\\displaystyle x^{5} - x + 15&\\displaystyle \\left(x^{2} + x + 3\\right) \\left(x^{3} - x^{2} - 2 x + 5\\right)\\\\\\displaystyle x^{5} - 9 x - 27&\\displaystyle \\left(x^{2} + 3 x + 3\\right) \\left(x^{3} - 3 x^{2} + 6 x - 9\\right)\\\\\\displaystyle x^{5} - 9 x + 27&\\displaystyle \\left(x^{2} - 3 x + 3\\right) \\left(x^{3} + 3 x^{2} + 6 x + 9\\right)\\\\\\displaystyle x^{5} - 5 x + 28&\\displaystyle \\left(x^{2} + x + 4\\right) \\left(x^{3} - x^{2} - 3 x + 7\\right)\\\\\\displaystyle x^{5} + 16 x + 32&\\displaystyle \\left(x^{2} + 2 x + 4\\right) \\left(x^{3} - 2 x^{2} + 8\\right)\\\\\\displaystyle x^{5} + 16 x - 32&\\displaystyle \\left(x^{2} - 2 x + 4\\right) \\left(x^{3} + 2 x^{2} - 8\\right)\\\\\\displaystyle x^{5} - 5 x^{2} + 2&\\displaystyle \\left(x^{2} - x - 1\\right) \\left(x^{3} + x^{2} + 2 x - 2\\right)\\\\\\displaystyle x^{5} - x^{2} + 4&\\displaystyle \\left(x^{2} + x + 2\\right) \\left(x^{3} - x^{2} - x + 2\\right)\\\\\\displaystyle x^{5} + 2 x^{2} + 4&\\displaystyle \\left(x^{2} - 2 x + 2\\right) \\left(x^{3} + 2 x^{2} + 2 x + 2\\right)\\\\\\displaystyle x^{5} - 3 x^{2} + 18&\\displaystyle \\left(x^{2} - 3 x + 3\\right) \\left(x^{3} + 3 x^{2} + 6 x + 6\\right)\\\\\\displaystyle x^{5} + x^{2} + 18&\\displaystyle \\left(x^{2} + x + 3\\right) \\left(x^{3} - x^{2} - 2 x + 6\\right)\\end{array}\\right]$\n"
      ],
      "text/plain": [
       "14×2 Matrix{Sym}:\n",
       "      x^5 + x + 1            (x^2 + x + 1)*(x^3 - x^2 + 1)\n",
       "    x^5 - 5*x + 3      (x^2 + x - 1)*(x^3 - x^2 + 2*x - 3)\n",
       "      x^5 + x + 6        (x^2 + x + 2)*(x^3 - x^2 - x + 3)\n",
       "     x^5 - x + 15      (x^2 + x + 3)*(x^3 - x^2 - 2*x + 5)\n",
       "   x^5 - 9*x - 27  (x^2 + 3*x + 3)*(x^3 - 3*x^2 + 6*x - 9)\n",
       "   x^5 - 9*x + 27  (x^2 - 3*x + 3)*(x^3 + 3*x^2 + 6*x + 9)\n",
       "   x^5 - 5*x + 28      (x^2 + x + 4)*(x^3 - x^2 - 3*x + 7)\n",
       "  x^5 + 16*x + 32        (x^2 + 2*x + 4)*(x^3 - 2*x^2 + 8)\n",
       "  x^5 + 16*x - 32        (x^2 - 2*x + 4)*(x^3 + 2*x^2 - 8)\n",
       "  x^5 - 5*x^2 + 2      (x^2 - x - 1)*(x^3 + x^2 + 2*x - 2)\n",
       "    x^5 - x^2 + 4        (x^2 + x + 2)*(x^3 - x^2 - x + 2)\n",
       "  x^5 + 2*x^2 + 4  (x^2 - 2*x + 2)*(x^3 + 2*x^2 + 2*x + 2)\n",
       " x^5 - 3*x^2 + 18  (x^2 - 3*x + 3)*(x^3 + 3*x^2 + 6*x + 6)\n",
       "   x^5 + x^2 + 18      (x^2 + x + 3)*(x^3 - x^2 - 2*x + 6)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Problems = [\n",
    "    x^5 + x + 1\n",
    "    x^5 - 5x + 3\n",
    "    x^5 + x + 6\n",
    "    x^5 - x + 15\n",
    "    x^5 - 9x - 27\n",
    "    x^5 - 9x + 27\n",
    "    x^5 - 5x + 28\n",
    "    x^5 + 16x + 32\n",
    "    x^5 + 16x - 32\n",
    "    x^5 - 5x^2 + 2\n",
    "    x^5 - x^2 + 4\n",
    "    x^5 + 2x^2 + 4\n",
    "    x^5 - 3x^2 + 18\n",
    "    x^5 + x^2 + 18\n",
    "]\n",
    "\n",
    "stack([f, f.factor()] for f in Problems; dims=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f06619e-07e9-4f55-a507-8066c3832df3",
   "metadata": {},
   "source": [
    "## 例1\n",
    "\n",
    "$g = x^2 + 3x - 2$, $h = (x+1)^3-2 = x^3+3x^2+3x-1$ の場合にどうなるか."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "2316fbfa-daa4-4475-8c0b-69867c8c795b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 3 x - 2$\n"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 3*x - 2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GG = x^2 + 3x - 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "08b30df4-848b-460e-8ddf-db2360d5da97",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} + 3 x^{2} + 3 x - 1$\n"
      ],
      "text/plain": [
       " 3      2          \n",
       "x  + 3*x  + 3*x - 1"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "HH = (x+1)^3-2 |> expand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "7d3cbc8c-13bf-48d2-b2e7-917158aa2694",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + 3 x - 2\\right) \\left(x^{3} + 3 x^{2} + 3 x - 1\\right)$\n"
      ],
      "text/plain": [
       "/ 2          \\ / 3      2          \\\n",
       "\\x  + 3*x - 2/*\\x  + 3*x  + 3*x - 1/"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF = GG*HH"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "4d111fb6-5c7b-470e-b198-948d6a9d51c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{5} + 6 x^{4} + 10 x^{3} + 2 x^{2} - 9 x + 2$\n"
      ],
      "text/plain": [
       " 5      4       3      2          \n",
       "x  + 6*x  + 10*x  + 2*x  - 9*x + 2"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF = expand(FF)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05131908-5323-4126-a7bf-2b3b8ec443c3",
   "metadata": {},
   "source": [
    "既存の函数を使えば因数分解は易しい."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "fe89deee-1831-4039-a19d-946026f83cdd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + 3 x - 2\\right) \\left(x^{3} + 3 x^{2} + 3 x - 1\\right)$\n"
      ],
      "text/plain": [
       "/ 2          \\ / 3      2          \\\n",
       "\\x  + 3*x - 2/*\\x  + 3*x  + 3*x - 1/"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(FF)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8bdfdccf-feaf-4ecf-991f-4f80625b929c",
   "metadata": {},
   "source": [
    "係数を $P,Q,R,S,T$ に代入する."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "75013cc9-6bc0-4654-aad4-93f4a6ac513c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle 6\\\\\\displaystyle 10\\\\\\displaystyle 2\\\\\\displaystyle -9\\\\\\displaystyle 2\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "6-element Vector{Sym}:\n",
       "  1\n",
       "  6\n",
       " 10\n",
       "  2\n",
       " -9\n",
       "  2"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_, P, Q, R, S, T = all_coeffs(FF, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "478a5a5b-0636-472b-9368-8d77ee5057bf",
   "metadata": {},
   "source": [
    "余りの定数項が0になるという条件."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "eb080670-4ef6-4f57-8581-cd8cfc396a9b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{3} - 6 a^{2} - 2 a b + 10 a + 6 b - 2 + \\frac{2}{b}$\n"
      ],
      "text/plain": [
       " 3      2                            2\n",
       "a  - 6*a  - 2*a*b + 10*a + 6*b - 2 + -\n",
       "                                     b"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem0 = eqrem0(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5690df89-8c11-4d2f-a222-7d7077ab9976",
   "metadata": {},
   "source": [
    "余りの1次の項が0になるという条件."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "61bd7ed1-9a74-46f1-8a6e-b5bb2c18369a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{4} - 6 a^{3} - 3 a^{2} b + 10 a^{2} + 12 a b - 2 a + b^{2} - 10 b - 9$\n"
      ],
      "text/plain": [
       " 4      3      2         2                   2           \n",
       "a  - 6*a  - 3*a *b + 10*a  + 12*a*b - 2*a + b  - 10*b - 9"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem1 = eqrem1(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac761886-dea6-465d-b580-d0ca1aa66e16",
   "metadata": {},
   "source": [
    "$be = 2$ となる整数の組 $(b, e)$ 全体について, 最下段の方程式が整数解を持つ場合を探す."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "cd9e057e-f04b-493c-8333-4ad0473e2049",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{rr}\\displaystyle 1&\\displaystyle a^{3} - 6 a^{2} + 8 a + 6\\\\\\displaystyle -1&\\displaystyle a^{3} - 6 a^{2} + 12 a - 10\\\\\\displaystyle 2&\\displaystyle a^{3} - 6 a^{2} + 6 a + 11\\\\\\displaystyle -2&\\displaystyle \\left(a - 3\\right) \\left(a^{2} - 3 a + 5\\right)\\end{array}\\right]$\n"
      ],
      "text/plain": [
       "4×2 Matrix{Sym}:\n",
       "  1    a^3 - 6*a^2 + 8*a + 6\n",
       " -1  a^3 - 6*a^2 + 12*a - 10\n",
       "  2   a^3 - 6*a^2 + 6*a + 11\n",
       " -2  (a - 3)*(a^2 - 3*a + 5)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor_for_all_divisors(Eqrem0, b, N(T))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e14a0e9b-c64b-4eb4-ab54-668d5c24165c",
   "metadata": {},
   "source": [
    "最後の $b=-2$ の場合にのみ整数解 $a=3$ が存在することがわかった."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "06cfd6a9-aec5-4c11-ae03-d4d19a6f0493",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2, 3)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB, AA = -2, 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c91fb73-f473-445c-9507-0e807c1063f2",
   "metadata": {},
   "source": [
    "この場合が実際に解になっていることの確認."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "134001f9-9e00-452a-9c25-0ddd114a4a8b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-a + p, a^2 - a*p - b + q, -a^3 + a^2*p + 2*a*b - a*q - b*p + r)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C, D, E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "c5203760-33ef-41bc-96c3-d660f9df4c08",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 3, -1)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CC, DD, EE = C(a=>AA, b=>BB, p=>P), D(a=>AA, b=>BB, p=>P, q=>Q), E(a=>AA, b=>BB, p=>P, q=>Q, r=>R, t=>T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "6aef550c-e718-420d-a089-855aa56b56ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem1(a=>AA, b=>BB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "500fe1d2-bd75-473a-a682-192a1d1e388b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + 3 x - 2\\right) \\left(x^{3} + 3 x^{2} + 3 x - 1\\right)$\n"
      ],
      "text/plain": [
       "/ 2          \\ / 3      2          \\\n",
       "\\x  + 3*x - 2/*\\x  + 3*x  + 3*x - 1/"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = (x^2 + AA*x + BB)*(x^3 + CC*x^2 + DD*x + EE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "5e1b5979-eaa3-49d8-9848-480793520d5b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol - FF |> simplify"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8fef769-ad5c-458d-8044-11981c9a19e5",
   "metadata": {},
   "source": [
    "## 例2\n",
    "\n",
    "$f = x^5 - x^4 - 1$ の因数分解."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "2c13915d-b8f7-47cf-911e-49b62a476104",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{5} - x^{4} - 1$\n"
      ],
      "text/plain": [
       " 5    4    \n",
       "x  - x  - 1"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF = x^5 - x^4 - 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3ca68f2-6f73-443b-b4c2-504d4d01d695",
   "metadata": {},
   "source": [
    "既存の函数を使えば易しい."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "1b820290-8f5b-4aa9-9663-e257af15b225",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} - x + 1\\right) \\left(x^{3} - x - 1\\right)$\n"
      ],
      "text/plain": [
       "/ 2        \\ / 3        \\\n",
       "\\x  - x + 1/*\\x  - x - 1/"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(FF)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7860708-6129-4869-a6c5-180a1d76c9d4",
   "metadata": {},
   "source": [
    "よくある解答例では天下り的に「$x^3$ を足して引く」という方法を使っている.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "x^5 - x^4 - 1\n",
    "&=\n",
    "x^5 - x^4 + x^3 - x^3 - 1\n",
    "\\\\ &=\n",
    "(x^5 - x^4 + x^3) - (x^3 + 1)\n",
    "\\\\ &=\n",
    "x^3(x^2-x+1) -(x+1)(x^2-x+1)\n",
    "\\\\ &=\n",
    "(x^2-x+1)(x^3-x-1).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "他の方法: $x^2-x+1 = 0$ (これの2つの解は虚数)のとき $x^3=-1$ なので, $x^5-x^4-1=-x^2+x-1=0$ となるので, $x^5 - x^4 - 1$ は $x^2-x+1$ で割り切れることがわかる."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6661d21-e08e-4b5e-bb15-08124f5ce5d1",
   "metadata": {},
   "source": [
    "以下では素朴な方法でも同じ結果が得られることを示す.\n",
    "\n",
    "係数を $P,Q,R,S,T$ に代入する."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "2c032626-3f4f-47aa-b699-874d51e6ffa4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle -1\\\\\\displaystyle 0\\\\\\displaystyle 0\\\\\\displaystyle 0\\\\\\displaystyle -1\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "6-element Vector{Sym}:\n",
       "  1\n",
       " -1\n",
       "  0\n",
       "  0\n",
       "  0\n",
       " -1"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_, P, Q, R, S, T = all_coeffs(FF, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e269c7d1-774f-475c-a330-650af29c4e45",
   "metadata": {},
   "source": [
    "余りの定数項が0になるという条件."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "b126ffc1-4670-4e2a-af44-333cfd9230f6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{3} + a^{2} - 2 a b - b - \\frac{1}{b}$\n"
      ],
      "text/plain": [
       " 3    2               1\n",
       "a  + a  - 2*a*b - b - -\n",
       "                      b"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem0 = eqrem0(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee7a2d75-1d4d-4f10-aefa-2303fb47ae82",
   "metadata": {},
   "source": [
    "余りの1次の項が0になるという条件."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "39c92b13-99aa-48ee-b339-0ce7db6213e8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{4} + a^{3} - 3 a^{2} b - 2 a b + b^{2}$\n"
      ],
      "text/plain": [
       " 4    3      2              2\n",
       "a  + a  - 3*a *b - 2*a*b + b "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem1 = eqrem1(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fafc1bd6-4267-49f9-b6ae-c4fdc6d508a0",
   "metadata": {},
   "source": [
    "$be=-1$ となる整数の組 $(b,d)$ の各々について, 最下段の方程式が整数解を持つかどうかを確認する."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "e3fe41b6-3ecc-4e22-91f4-0bd626e41fbf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{rr}\\displaystyle 1&\\displaystyle \\left(a + 1\\right) \\left(a^{2} - 2\\right)\\\\\\displaystyle -1&\\displaystyle \\left(a + 1\\right) \\left(a^{2} + 2\\right)\\end{array}\\right]$\n"
      ],
      "text/plain": [
       "2×2 Matrix{Sym}:\n",
       "  1  (a + 1)*(a^2 - 2)\n",
       " -1  (a + 1)*(a^2 + 2)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor_for_all_divisors(Eqrem0, b, N(T))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfb90943-6c64-4dc1-866f-342031914ac6",
   "metadata": {},
   "source": [
    "$b=\\pm 1$ の両方の場合に整数解 $a=-1$ を持つことがわかった.\n",
    "\n",
    "これらが下から2段目の方程式も満たしているかを確認する."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "891f232a-d4ca-4679-bd09-f6cd47ee419e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB, AA = 1, -1\n",
    "Eqrem1(a=>AA, b=>BB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "0f71e001-416e-487b-bbf4-eb00de2e185b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$\n"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB, AA = -1, -1\n",
    "Eqrem1(a=>AA, b=>BB)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d2ed6509-4426-4f05-89f6-e35c479b9227",
   "metadata": {},
   "source": [
    "$(a, b) = -1, 1$ の場合のみが解になっていることがわかった."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22b7194d-f43c-44ce-a8c9-137901f11d12",
   "metadata": {},
   "source": [
    "以下は検算である."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "80feb09b-0858-4bcf-8987-3256012df508",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, -1, -1)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AA, BB = -1, 1\n",
    "CC, DD, EE = C(a=>AA, b=>BB, p=>P), D(a=>AA, b=>BB, p=>P, q=>Q), E(a=>AA, b=>BB, p=>P, q=>Q, r=>R, t=>T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "3565b1b0-d959-4d78-8660-59194170a5be",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} - x + 1\\right) \\left(x^{3} - x - 1\\right)$\n"
      ],
      "text/plain": [
       "/ 2        \\ / 3        \\\n",
       "\\x  - x + 1/*\\x  - x - 1/"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = (x^2 + AA*x + BB)*(x^3 + CC*x^2 + DD*x + EE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "fd159a47-ca9d-42f0-a78d-bfbd22762938",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol - FF |> simplify"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d1d35cf7-1b60-4e83-bcb0-00981bdc689d",
   "metadata": {},
   "source": [
    "## 例3\n",
    "\n",
    "$f = x^5 + 16x + 32$ の因数分解."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "24af85b2-a6d4-4154-837e-32afd11d670d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{5} + 16 x + 32$\n"
      ],
      "text/plain": [
       " 5            \n",
       "x  + 16*x + 32"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF = x^5 + 16x + 32"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2395e598-0fdc-40e0-8176-3dd313695694",
   "metadata": {},
   "source": [
    "既存の函数を使えば易しい."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "0ca2ad7b-95fb-486f-86f5-3054d941782f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + 2 x + 4\\right) \\left(x^{3} - 2 x^{2} + 8\\right)$\n"
      ],
      "text/plain": [
       "/ 2          \\ / 3      2    \\\n",
       "\\x  + 2*x + 4/*\\x  - 2*x  + 8/"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(FF)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d31e402b-5b27-4db5-89ed-d75fd822660e",
   "metadata": {},
   "source": [
    "$f$ に $x=2z$ を代入して得られる\n",
    "\n",
    "$$\n",
    "f|_{x=2z} = 32(z^5+z+1)\n",
    "$$\n",
    "\n",
    "は $z^2+z+1=0$ (これの2つの解は虚数)のとき, $z^3=1$, $z^5=z^2$ となるので $0$ になる.\n",
    "\n",
    "ゆえに $z^2+z+1$ に $z=x/2$ を代入したものの4倍\n",
    "\n",
    "$$\n",
    "x^2 + 2x + 4\n",
    "$$\n",
    "\n",
    "で $f$ は割り切れる."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "07e8b6d4-c18f-4585-9057-a4a5428506cd",
   "metadata": {},
   "source": [
    "以下では素朴な方法でも同じ結果が得られることを示そう.\n",
    "\n",
    "係数を $P,Q,R,S,T$ に代入する."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "65d453b0-e174-457d-a9bf-4026368218b9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle 0\\\\\\displaystyle 0\\\\\\displaystyle 0\\\\\\displaystyle 16\\\\\\displaystyle 32\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "6-element Vector{Sym}:\n",
       "  1\n",
       "  0\n",
       "  0\n",
       "  0\n",
       " 16\n",
       " 32"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_, P, Q, R, S, T = all_coeffs(FF, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22b021b7-7af2-43d5-8bb2-ac8234726640",
   "metadata": {},
   "source": [
    "余りの定数項が0になるという条件."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "65ef985c-67c2-4a18-893a-a39c1702019e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{3} - 2 a b + \\frac{32}{b}$\n"
      ],
      "text/plain": [
       " 3           32\n",
       "a  - 2*a*b + --\n",
       "             b "
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem0 = eqrem0(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9879cc28-5968-4fdd-8464-e8533e2e29ce",
   "metadata": {},
   "source": [
    "余りの1次の項が0になるという条件."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "6efca41f-970f-4ea1-9753-2a3456d94d2a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{4} - 3 a^{2} b + b^{2} + 16$\n"
      ],
      "text/plain": [
       " 4      2      2     \n",
       "a  - 3*a *b + b  + 16"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem1 = eqrem1(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45a9a6a2-2910-4229-bfe1-4ad8f6074220",
   "metadata": {},
   "source": [
    "$b$ として $32$ の約数全体を動かして, `eqrem0` を0にする整数 $a$ が存在する場合を探す."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "70fd2fcd-3553-4699-b303-3988aa628756",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{rr}\\displaystyle 1&\\displaystyle a^{3} - 2 a + 32\\\\\\displaystyle -1&\\displaystyle a^{3} + 2 a - 32\\\\\\displaystyle 2&\\displaystyle a^{3} - 4 a + 16\\\\\\displaystyle -2&\\displaystyle \\left(a - 2\\right) \\left(a^{2} + 2 a + 8\\right)\\\\\\displaystyle 4&\\displaystyle \\left(a - 2\\right) \\left(a^{2} + 2 a - 4\\right)\\\\\\displaystyle -4&\\displaystyle a^{3} + 8 a - 8\\\\\\displaystyle 8&\\displaystyle a^{3} - 16 a + 4\\\\\\displaystyle -8&\\displaystyle a^{3} + 16 a - 4\\\\\\displaystyle 16&\\displaystyle a^{3} - 32 a + 2\\\\\\displaystyle -16&\\displaystyle a^{3} + 32 a - 2\\\\\\displaystyle 32&\\displaystyle a^{3} - 64 a + 1\\\\\\displaystyle -32&\\displaystyle a^{3} + 64 a - 1\\end{array}\\right]$\n"
      ],
      "text/plain": [
       "12×2 Matrix{Sym}:\n",
       "   1           a^3 - 2*a + 32\n",
       "  -1           a^3 + 2*a - 32\n",
       "   2           a^3 - 4*a + 16\n",
       "  -2  (a - 2)*(a^2 + 2*a + 8)\n",
       "   4  (a - 2)*(a^2 + 2*a - 4)\n",
       "  -4            a^3 + 8*a - 8\n",
       "   8           a^3 - 16*a + 4\n",
       "  -8           a^3 + 16*a - 4\n",
       "  16           a^3 - 32*a + 2\n",
       " -16           a^3 + 32*a - 2\n",
       "  32           a^3 - 64*a + 1\n",
       " -32           a^3 + 64*a - 1"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor_for_all_divisors(Eqrem0, b, N(T))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47e8642d-b78a-4c97-ab42-5a2324e6aa36",
   "metadata": {},
   "source": [
    "方程式 `eqrem0 = 0` は $b=4,-2$ のときにのみ整数解 $a=2$ を持つことがわかった."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "3e58b87c-4cbb-4a15-a1c3-1009bc73dbc9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB, AA = 4, 2\n",
    "Eqrem1(a=>AA, b=>BB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "36fcd873-8dc7-45f1-bece-d9eb6f007239",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 60$\n"
      ],
      "text/plain": [
       "60"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB, AA = -2, 2\n",
    "Eqrem1(a=>AA, b=>BB)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8cb3b847-8cca-49c5-a841-504994da80f3",
   "metadata": {},
   "source": [
    "方程式 `eqrem0 = eqrem1 = 0` の整数解 $a = 2$, $b=4$ が見付かった."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "1acce335-5e2a-43d8-b173-4920666a4994",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2, 0, 8)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AA, BB = 2, 4\n",
    "CC, DD, EE = C(a=>AA, b=>BB, p=>P), D(a=>AA, b=>BB, p=>P, q=>Q), E(a=>AA, b=>BB, p=>P, q=>Q, r=>R, t=>T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "a457dfd8-b6fc-407e-9236-0e3276175193",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + 2 x + 4\\right) \\left(x^{3} - 2 x^{2} + 8\\right)$\n"
      ],
      "text/plain": [
       "/ 2          \\ / 3      2    \\\n",
       "\\x  + 2*x + 4/*\\x  - 2*x  + 8/"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = (x^2 + AA*x + BB)*(x^3 + CC*x^2 + DD*x + EE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "ac373687-98c3-404e-9a1a-61dcdb8eff04",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol - FF |> simplify"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32c5229d-eb72-4151-98c8-03142b34e0b1",
   "metadata": {},
   "source": [
    "## 例4\n",
    "\n",
    "$f = x^5 + 3x^4 - 2x^3 - 2x^2 - 6x + 4$ の因数分解"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "1757f818-64d1-4f79-a452-a395179e6ac6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{5} + 3 x^{4} - 2 x^{3} - 2 x^{2} - 6 x + 4$\n"
      ],
      "text/plain": [
       " 5      4      3      2          \n",
       "x  + 3*x  - 2*x  - 2*x  - 6*x + 4"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF = x^5 + 3x^4 - 2x^3 - 2x^2 - 6x + 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "047ff974-a50a-47c2-8fd5-799e070f96c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{3} - 2\\right) \\left(x^{2} + 3 x - 2\\right)$\n"
      ],
      "text/plain": [
       "/ 3    \\ / 2          \\\n",
       "\\x  - 2/*\\x  + 3*x - 2/"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(FF)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a136eb0-50ef-4583-9b58-45dd98d02924",
   "metadata": {},
   "source": [
    "$f$ が $x^k + a$ 型の因子を持つと決め打ちできるなら, そのように $f$ を整理すれば容易に因数分解できる.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "x^5 + 3x^4 - 2x^3 - 2x^2 - 6x + 4\n",
    "&=\n",
    "(x^5 + 3x^4 - 2x^3) - (2x^2 + 6x - 4)\n",
    "\\\\ &=\n",
    "x^3(x^2 + 3x - 2) - 2(x^2 + 3x - 2)\n",
    "\\\\ &=\n",
    "(x^3-2)(x^2 + 3x - 2).\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "7fd6db4b-b273-4d9b-a90d-f2a16348d734",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{3} - 2\\right) \\left(x^{2} + 3 x - 2\\right)$\n"
      ],
      "text/plain": [
       "/ 3    \\ / 2          \\\n",
       "\\x  - 2/*\\x  + 3*x - 2/"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(FF)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "d8e7adcc-88a6-445a-bc14-df0642840865",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle 3\\\\\\displaystyle -2\\\\\\displaystyle -2\\\\\\displaystyle -6\\\\\\displaystyle 4\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "6-element Vector{Sym}:\n",
       "  1\n",
       "  3\n",
       " -2\n",
       " -2\n",
       " -6\n",
       "  4"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_, P, Q, R, S, T = all_coeffs(FF, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "134a281b-876f-4a1c-9ee7-0f5c0cb4f804",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{3} - a^{2} p - 2 a b + a q + b p - r + \\frac{t}{b}$\n"
      ],
      "text/plain": [
       " 3    2                             t\n",
       "a  - a *p - 2*a*b + a*q + b*p - r + -\n",
       "                                    b"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqrem0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "1b990dda-d211-4c93-a7e0-03ae6d98d447",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{3} - 3 a^{2} - 2 a b - 2 a + 3 b + 2 + \\frac{4}{b}$\n"
      ],
      "text/plain": [
       " 3      2                           4\n",
       "a  - 3*a  - 2*a*b - 2*a + 3*b + 2 + -\n",
       "                                    b"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem0 = eqrem0(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "51b044d6-6ec2-4800-83eb-2237379b9b56",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{4} - a^{3} p - 3 a^{2} b + a^{2} q + 2 a b p - a r + b^{2} - b q + s$\n"
      ],
      "text/plain": [
       " 4    3        2      2                      2          \n",
       "a  - a *p - 3*a *b + a *q + 2*a*b*p - a*r + b  - b*q + s"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqrem1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "9729f33c-3bba-4211-889b-4196f86628a9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a^{4} - 3 a^{3} - 3 a^{2} b - 2 a^{2} + 6 a b + 2 a + b^{2} + 2 b - 6$\n"
      ],
      "text/plain": [
       " 4      3      2        2                  2          \n",
       "a  - 3*a  - 3*a *b - 2*a  + 6*a*b + 2*a + b  + 2*b - 6"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eqrem1 = eqrem1(p=>P, q=>Q, r=>R, s=>S, t=>T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "086ad167-0079-44e6-b6ee-fe6d5262d1a4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{rr}\\displaystyle 1&\\displaystyle a^{3} - 3 a^{2} - 4 a + 9\\\\\\displaystyle -1&\\displaystyle a^{3} - 3 a^{2} - 5\\\\\\displaystyle 2&\\displaystyle a^{3} - 3 a^{2} - 6 a + 10\\\\\\displaystyle -2&\\displaystyle \\left(a - 3\\right) \\left(a^{2} + 2\\right)\\\\\\displaystyle 4&\\displaystyle a^{3} - 3 a^{2} - 10 a + 15\\\\\\displaystyle -4&\\displaystyle a^{3} - 3 a^{2} + 6 a - 11\\end{array}\\right]$\n"
      ],
      "text/plain": [
       "6×2 Matrix{Sym}:\n",
       "  1    a^3 - 3*a^2 - 4*a + 9\n",
       " -1          a^3 - 3*a^2 - 5\n",
       "  2   a^3 - 3*a^2 - 6*a + 10\n",
       " -2        (a - 3)*(a^2 + 2)\n",
       "  4  a^3 - 3*a^2 - 10*a + 15\n",
       " -4   a^3 - 3*a^2 + 6*a - 11"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor_for_all_divisors(Eqrem0, b, N(T))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "0616bd8a-0758-450c-8a6d-3894cd5acdf5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB, AA = -2, 3\n",
    "Eqrem1(a=>AA, b=>BB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "abc91ac0-82a8-426f-84a9-47b2906ded58",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0, -2)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AA, BB = 3, -2\n",
    "CC, DD, EE = C(a=>AA, b=>BB, p=>P), D(a=>AA, b=>BB, p=>P, q=>Q), E(a=>AA, b=>BB, p=>P, q=>Q, r=>R, t=>T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "bf190e8c-54cb-4436-a202-f89d0b128015",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{3} - 2\\right) \\left(x^{2} + 3 x - 2\\right)$\n"
      ],
      "text/plain": [
       "/ 3    \\ / 2          \\\n",
       "\\x  - 2/*\\x  + 3*x - 2/"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = (x^2 + AA*x + BB)*(x^3 + CC*x^2 + DD*x + EE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "19c1391f-a40d-41aa-8d3f-6721ba3962f9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$\n"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol - FF |> simplify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0edf973a-9e6d-47ff-ac54-1613a7a10948",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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