m81336 Multiply Polynomials By the end of this section, you will be able to: Multiply a polynomial by a monomial Multiply a binomial by a binomial Multiply a trinomial by a binomial 6eddcee7-b953-46db-92b4-db3be81acaed Before you get started, take this readiness quiz. Distribute: 2(x+3). If you missed the problem, review . 2x+6 Distribute: −11(4−3a). If you missed the problem, review . −44+33a Combine like terms: x2+9x+7x+63. If you missed the problem, review . x2+16x+63

In Distributive Property you learned to use the Distributive Property to simplify expressions such as 2(x−3). You multiplied both terms in the parentheses, xand3, by 2, to get 2x−6. With this chapter's new vocabulary, you can say you were multiplying a binomial, x−3, by a monomial, 2. Multiplying a binomial by a monomial is nothing new for you! Multiply: 3(x+7). 3(x+7) Distribute. 3·x+3·7 Simplify. 3x+21 Multiply: 6(x+8). 6x + 48 Multiply: 2(y+12). 2y + 24 Multiply: x(x−8). Distribute. Simplify. Multiply: y(y−9). y² − 9y Multiply: p(p−13). p² − 13p Multiply: 10x(4x+y). Distribute. Simplify. Multiply: 8x(x+3y). 8x² + 24xy Multiply: 3r(6r+s). 18r² + 3rs Multiplying a monomial by a trinomial works in much the same way. Multiply: −2x(5x2+7x−3). −2x(5x2+7x−3) Distribute. −2x⋅5x2+(−2x)⋅7x−(−2x)⋅3 Simplify. −10x3−14x2+6x Multiply: −4y(8y2+5y−9). −32y³ − 20y² + 36y Multiply: −6x(9x2+x−1). −54x³ − 6x² + 6x Multiply: 4y3(y2−8y+1). 4y3(y2−8y+1) Distribute. 4y3⋅y2−4y3⋅8y+4y3⋅1 Simplify. 4y5−32y4+4y3 Multiply: 3x2(4x2−3x+9). 12x⁴ − 9x³ + 27x² Multiply: 8y2(3y2−2y−4). 24y⁴ − 16y³ − 32y² Now we will have the monomial as the second factor. Multiply: (x+3)p. (x+3)p Distribute. x⋅p+3⋅p Simplify. xp+3p Multiply: (x+8)p. xp + 8p Multiply: (a+4)p. ap + 4p

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

We will start by using the Distributive Property. Look again at . We distributed the p to get What if we have (x+7) instead of p? Distribute (x+7). Distribute again. x2+7x+3x+21 Combine like terms. x2+10x+21Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications. Be careful to distinguish between a sum and a product. SumProductx+xx·x2xx2combine like termsadd exponents of like bases Multiply: (x+6)(x+8). (x+6)(x+8) Distribute (x+8). Distribute again. x2+8x+6x+48 Simplify. x2+14x+48 Multiply: (x+8)(x+9). x² + 17x + 72 Multiply: (a+4)(a+5). a² + 9a + 20 Now we'll see how to multiply binomials where the variable has a coefficient. Multiply: (2x+9)(3x+4). (2x+9)(3x+4) Distribute. (3x+4) Distribute again. 6x2+8x+27x+36 Simplify. 6x2+35x+36 Multiply: (5x+9)(4x+3). 20x² + 51x + 27 Multiply: (10m+9)(8m+7). 80m² + 142m + 63 In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct. Multiply: (4y+3)(6y−5). (4y+3)(6y−5) Distribute. Distribute again. 24y2−20y+18y−15 Simplify. 24y2−2y−15 Multiply: (7y+1)(8y−3). 56y² − 13y − 3 Multiply: (3x+2)(5x−8). 15x² − 14x − 16 Up to this point, the product of two binomials has been a trinomial. This is not always the case. Multiply: (x+2)(x−y). Distribute. Distribute again. Simplify. There are no like terms to combine. Multiply: (x+5)(x−y). x² − xy + 5x − 5y Multiply: (x+2y)(x−1). x² − x + 2xy − 2y

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms. (x+2)(x−y) x2−xy+2x−2y Where did the first term, x2, come from? It is the product of xandx, the first terms in (x+2)and(x−y). The next term, −xy, is the product of xand−y, the two outer terms. The third term, +2x, is the product of 2andx, the two inner terms. And the last term, −2y, came from multiplying the two last terms. We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials. Let's look at (x+3)(x+7) again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.

Multiply using the FOIL method: (x+6)(x+9). Step 1: Multiply the First terms. Step 2: Multiply the Outer terms. Step 3: Multiply the Inner terms. Step 4: Multiply the Last terms. Step 5: Combine like terms, when possible. Multiply using the FOIL method: (x+7)(x+8). x² + 15x + 56 Multiply using the FOIL method: (y+14)(y+2). y² + 16y + 28 We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!Multiply the First terms. Multiply the Outer terms. Multiply the Inner terms. Multiply the Last terms. Combine like terms, when possible. Multiply: (y−8)(y+6). Step 1: Multiply the First terms. Step 2: Multiply the Outer terms. Step 3: Multiply the Inner terms. Step 4: Multiply the Last terms. Step 5: Combine like terms Multiply: (y−3)(y+8). y² + 5y − 24 Multiply: (q−4)(q+5). q² + q − 20 Multiply: (2a+3)(3a−1). Multiply the First terms. Multiply the Outer terms. Multiply the Inner terms. Multiply the Last terms. Combine like terms. Multiply: (4a+9)(5a−2). 20a² + 37a − 18 Multiply: (7x+4)(7x−8). 49x² − 28x − 32 Multiply: (5x−y)(2x−7). Multiply the First terms. Multiply the Outer terms. Multiply the Inner terms. Multiply the Last terms. Combine like terms. There are none. Multiply: (12x−y)(x−5). 12x² − 60x − xy + 5y Multiply: (6a−b)(2a−9). 12a² − 54a − 2ab + 9b

The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers. You start by multiplying 23 by 6 to get 138.Then you multiply 23 by 4, lining up the partial product in the correct columns.Last, you add the partial products.Now we'll apply this same method to multiply two binomials. Multiply using the vertical method: (5x−1)(2x−7). It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied 23(46). Multiply 2x−7 by −1. Multiply 2x−7 by 5x. Add like terms. Notice the partial products are the same as the terms in the FOIL method. Multiply using the vertical method: (4m−9)(3m−7). 12m² − 55m + 63 Multiply using the vertical method: (6n−5)(7n−2). 42n² − 47n + 10 We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.To multiply binomials, use the: Distributive Property FOIL Method Vertical Method Remember, FOIL only works when multiplying two binomials.

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property. Multiply using the Distributive Property: (x+3)(2x2−5x+8). Distribute. Multiply. 2x3−5x2+8x+6x2−15x+24 Combine like terms. 2x3+x2−7x+24 Multiply using the Distributive Property: (y−1)(y2−7y+2). y³ − 8y² + 9y − 2 Multiply using the Distributive Property: (x+2)(3x2−4x+5). 3x³ + 2x² − 3x + 10 Now let's do this same multiplication using the Vertical Method. Multiply using the Vertical Method: (x+3)(2x2−5x+8). It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way. Multiply (2x2−5x+8) by 3. Multiply (2x2−5x+8) by x. Add like terms. Multiply using the Vertical Method: (y−1)(y2−7y+2). y³ − 8y² + 9y − 2 Multiply using the Vertical Method: (x+2)(3x2−4x+5). 3x³ + 2x² − 3x + 10 Multiply Monomials Multiply Polynomials Multiply Polynomials 2 Multiply Polynomials Review Multiply Polynomials Using the Distributive Property Multiply Binomials

Use the FOIL method for multiplying two binomials. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms. Step 5. Combine like terms, when possible. Multiplying Two Binomials: To multiply binomials, use the: Distributive Property FOIL Method Vertical Method Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the: Distributive Property Vertical Method

Multiply a Polynomial by a Monomial In the following exercises, multiply. 4(x+10) 4x + 40 6(y+8) 15(r−24) 15r − 360 12(v−30) −3(m+11) −3m − 33 −4(p+15) −8(z−5) −8z + 40 −3(x−9) u(u+5) u² + 5u q(q+7) n(n2−3n) n³ − 3n² s(s2−6s) 12x(x−10) 12x² − 120x 9m(m−11) −9a(3a+5) −27a² − 45a −4p(2p+7) 6x(4x+y) 24x² + 6xy 5a(9a+b) 5p(11p−5q) 55p² − 25pq 12u(3u−4v) 3(v2+10v+25) 3v² + 30v + 75 6(x2+8x+16) 2n(4n2−4n+1) 8n³ − 8n² + 2n 3r(2r2−6r+2) −8y(y2+2y−15) −8y³ − 16y² + 120y −5m(m2+3m−18) 5q3(q2−2q+6) 5q⁵ − 10q⁴ + 30q³ 9r3(r2−3r+5) −4z2(3z2+12z−1) −12z⁴ − 48z³ + 4z² −3x2(7x2+10x−1) (2y−9)y 2y² − 9y (8b−1)b (w−6)·8 8w − 48 (k−4)·5 Multiply a Binomial by a Binomial In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical method (x+4)(x+6) x² + 10x + 24 (u+8)(u+2) (n+12)(n−3) n² + 9n − 36 (y+3)(y−9) In the following exercises, multiply the following binomials. Use any method. (y+8)(y+3) y² + 11y + 24 (x+5)(x+9) (a+6)(a+16) a² + 22a + 96 (q+8)(q+12) (u−5)(u−9) u² − 14u + 45 (r−6)(r−2) (z−10)(z−22) z² − 32z + 220 (b−5)(b−24) (x−4)(x+7) x² + 3x − 28 (s−3)(s+8) (v+12)(v−5) v² + 7v − 60 (d+15)(d−4) (6n+5)(n+1) 6n² + 11n + 5 (7y+1)(y+3) (2m−9)(10m+1) 20m² − 88m − 9 (5r−4)(12r+1) (4c−1)(4c+1) 16c² − 1 (8n−1)(8n+1) (3u−8)(5u−14) 15u² − 82u + 112 (2q−5)(7q−11) (a+b)(2a+3b) 2a² + 5ab + 3b² (r+s)(3r+2s) (5x−y)(x−4) 5x² − 20x − xy + 4y (4z−y)(z−6) Multiply a Trinomial by a Binomial In the following exercises, multiply using ⓐ the Distributive Property and ⓑ the Vertical Method. (u+4)(u2+3u+2) u³ + 7u² + 14u + 8 (x+5)(x2+8x+3) (a+10)(3a2+a−5) 3a³ + 31a² + 5a − 50 (n+8)(4n2+n−7) In the following exercises, multiply. Use either method. (y−6)(y2−10y+9) y³ − 16y² + 69y − 54 (k−3)(k2−8k+7) (2x+1)(x2−5x−6) 2x³ − 9x² − 17x − 6 (5v+1)(v2−6v−10)

Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 13 times 15. Think of 13 as 10+3 and 15 as 10+5. ⓐ Multiply (10+3)(10+5) by the FOIL method. ⓑ Multiply 13·15 without using a calculator. ⓒ Which way is easier for you? Why? ⓐ 195ⓑ 195ⓒ Answers will vary. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as 20−2 and 17 as 20−3. ⓐ Multiply (20−2)(20−3) by the FOIL method. ⓑ Multiply 18·17 without using a calculator. ⓒ Which way is easier for you? Why?

Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why? Answers will vary. Which method do you prefer to use when multiplying a trinomial by a binomial—the Distributive Property or the Vertical Method? Why?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?