Multiplying Polynomials 6-5 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Objective Multiply polynomials.

To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

Example 1: Multiplying Monomials A. (6y3)(3y5) (6y3)(3y5) Group factors with like bases together. (6 3)(y3 y5)  18y8 Multiply. B. (3mn2) (9m2n) (3mn2)(9m2n) Group factors with like bases together. (3 9)(m m2)(n2  n)  27m3n3 Multiply.

Group factors with like bases together. (3x3)(6x2) Check It Out! Example 1 Multiply. a. (3x3)(6x2) Group factors with like bases together. (3x3)(6x2) (3 6)(x3 x2)  Multiply. 18x5 b. (2r2t)(5t3) Group factors with like bases together. (2r2t)(5t3) (2 5)(r2)(t3 t)  Multiply. 10r2t4

Check It Out! Example 1 Continued Multiply. æ 1 ö ( ) ( ) c. x y 2 12 x z 3 2 4 5 ç ÷ y z è 3 ø ( ) æ ç è 4 5 2 1 12 3 x z y ö ÷ ø Group factors with like bases together. ( ) æ ç è 3 2 4 5 1 12 z x x y y ö ÷ ø • • Multiply. • • 7 5 4 x y z

To multiply a polynomial by a monomial, use the Distributive Property.

Example 2A: Multiplying a Polynomial by a Monomial 4(3x2 + 4x – 8) Distribute 4. 4(3x2 + 4x – 8) (4)3x2 +(4)4x – (4)8 Multiply. 12x2 + 16x – 32

Check It Out! Example 2 Multiply. b. 3ab(5a2 + b) Distribute 3ab. 3ab(5a2 + b) (3ab)(5a2) + (3ab)(b) Group like bases together. (3  5)(a  a2)(b) + (3)(a)(b  b) 15a3b + 3ab2 Multiply.

Another method for multiplying binomials is called the FOIL method. 1. Multiply the First terms. (x + 3)(x + 2) x x = x2  O 2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x  I 3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x  L 4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6  (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6 F O I L

Example 3A: Multiplying Binomials (s + 4)(s – 2) (s + 4)(s – 2) s(s – 2) + 4(s – 2) Distribute. s(s) + s(–2) + 4(s) + 4(–2) Distribute again. s2 – 2s + 4s – 8 Multiply. s2 + 2s – 8 Combine like terms.

Example 3B: Multiplying Binomials Write as a product of two binomials. (x – 4)2 (x – 4)(x – 4) Use the FOIL method. (x x) + (x (–4)) + (–4  x) + (–4  (–4))  x2 – 4x – 4x + 16 Multiply. x2 – 8x + 16 Combine like terms.

Example 3C: Multiplying Binomials (8m2 – n)(m2 – 3n) Use the FOIL method. 8m2(m2) + 8m2(–3n) – n(m2) – n(–3n) 8m4 – 24m2n – m2n + 3n2 Multiply. 8m4 – 25m2n + 3n2 Combine like terms.

In the expression (x + 5)2, the base is (x + 5) In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5) Helpful Hint

Check It Out! Example 3a Multiply. (a + 3)(a – 4) (a + 3)(a – 4) a(a – 4)+3(a – 4) Distribute. a(a) + a(–4) + 3(a) + 3(–4) Distribute again. a2 – 4a + 3a – 12 Multiply. a2 – a – 12 Combine like terms.

Write as a product of two binomials. (x – 3)2 Check It Out! Example 3b Multiply. Write as a product of two binomials. (x – 3)2 (x – 3)(x – 3) Use the FOIL method. (x x) + (x(–3)) + (–3  x)+ (–3)(–3) ● x2 – 3x – 3x + 9 Multiply. x2 – 6x + 9 Combine like terms.

Check It Out! Example 3c Multiply. (2a – b2)(a + 4b2) (2a – b2)(a + 4b2) Use the FOIL method. 2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2) 2a2 + 8ab2 – ab2 – 4b4 Multiply. 2a2 + 7ab2 – 4b4 Combine like terms.

To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6): (5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6) = 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6) = 10x3 + 50x2 – 30x + 6x2 + 30x – 18 = 10x3 + 56x2 – 18

You can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x2 + 10x – 6) and width (5x + 3): 2x2 +10x –6 10x3 50x2 –30x 30x 6x2 –18 5x +3 Write the product of the monomials in each row and column: To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary. 10x3 + 6x2 + 50x2 + 30x – 30x – 18 10x3 + 56x2 – 18

Another method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers. Multiply each term in the top polynomial by 3. 2x2 + 10x – 6 5x + 3  Multiply each term in the top polynomial by 5x, and align like terms. 6x2 + 30x – 18 + 10x3 + 50x2 – 30x Combine like terms by adding vertically. 10x3 + 56x2 + 0x – 18 10x3 + 56x2 + – 18 Simplify.

Example 4A: Multiplying Polynomials (x – 5)(x2 + 4x – 6) (x – 5 )(x2 + 4x – 6) Distribute x. x(x2 + 4x – 6) – 5(x2 + 4x – 6) Distribute x again. x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6) x3 + 4x2 – 5x2 – 6x – 20x + 30 Simplify. x3 – x2 – 26x + 30 Combine like terms.

Example 4B: Multiplying Polynomials (2x – 5)(–4x2 – 10x + 3) Multiply each term in the top polynomial by –5. (2x – 5)(–4x2 – 10x + 3) Multiply each term in the top polynomial by 2x, and align like terms. –4x2 – 10x + 3 2x – 5 x 20x2 + 50x – 15 + –8x3 – 20x2 + 6x Combine like terms by adding vertically. –8x3 + 56x – 15

Example 4C: Multiplying Polynomials Write as the product of three binomials. [x · x + x(3) + 3(x) + (3)(3)] [x(x+3) + 3(x+3)](x + 3) Use the FOIL method on the first two factors. (x2 + 3x + 3x + 9)(x + 3) Multiply. (x2 + 6x + 9)(x + 3) Combine like terms.

Example 4C: Multiplying Polynomials Continued Use the Commutative Property of Multiplication. (x + 3)(x2 + 6x + 9) x(x2 + 6x + 9) + 3(x2 + 6x + 9) Distribute. x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9) Distribute again. x3 + 6x2 + 9x + 3x2 + 18x + 27 Combine like terms. x3 + 9x2 + 27x + 27

Example 4D: Multiplying Polynomials (3x + 1)(x3 + 4x2 – 7) Write the product of the monomials in each row and column. x3 –4x2 –7 3x 3x4 –12x3 –21x +1 –4x2 x3 –7 Add all terms inside the rectangle. 3x4 – 12x3 + x3 – 4x2 – 21x – 7 3x4 – 11x3 – 4x2 – 21x – 7 Combine like terms.

A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or 6 terms before simplifying. Helpful Hint 

Check It Out! Example 4a Multiply. (x + 3)(x2 – 4x + 6) (x + 3 )(x2 – 4x + 6) Distribute. x(x2 – 4x + 6) + 3(x2 – 4x + 6) Distribute again. x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6) x3 – 4x2 + 3x2 +6x – 12x + 18 Simplify. x3 – x2 – 6x + 18 Combine like terms.

Check It Out! Example 4b Multiply. (3x + 2)(x2 – 2x + 5) Multiply each term in the top polynomial by 2. (3x + 2)(x2 – 2x + 5) Multiply each term in the top polynomial by 3x, and align like terms. x2 – 2x + 5 3x + 2  2x2 – 4x + 10 + 3x3 – 6x2 + 15x Combine like terms by adding vertically. 3x3 – 4x2 + 11x + 10

Write the formula for the area of a rectangle. Example 5: Application The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. a. Write a polynomial that represents the area of the base of the prism. A = l  w A = l w  Write the formula for the area of a rectangle. Substitute h – 3 for w and h + 4 for l. A = (h + 4)(h – 3) A = h2 + 4h – 3h – 12 Multiply. A = h2 + h – 12 Combine like terms. The area is represented by h2 + h – 12.

Example 5: Application Continued The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. b. Find the area of the base when the height is 5 ft. A = h2 + h – 12 Write the formula for the area the base of the prism. A = h2 + h – 12 A = 52 + 5 – 12 Substitute 5 for h. A = 25 + 5 – 12 Simplify. A = 18 Combine terms. The area is 18 square feet.

The length of a rectangle is 4 meters shorter than its width. Check It Out! Example 5 The length of a rectangle is 4 meters shorter than its width. a. Write a polynomial that represents the area of the rectangle. A = l w  A = l w Write the formula for the area of a rectangle. Substitute x – 4 for l and x for w. A = x(x – 4) A = x2 – 4x Multiply. The area is represented by x2 – 4x.

Check It Out! Example 5 Continued The length of a rectangle is 4 meters shorter than its width. b. Find the area of a rectangle when the width is 6 meters. A = x2 – 4x Write the formula for the area of a rectangle whose length is 4 meters shorter than width . A = x2 – 4x A = 62 – 4  6 Substitute 6 for x. A = 36 – 24 Simplify. A = 12 Combine terms. The area is 12 square meters.

Lesson Quiz: Part I Multiply. 1. (6s2t2)(3st) 2. 4xy2(x + y) 3. (x + 2)(x – 8) 4. (2x – 7)(x2 + 3x – 4) 5. 6mn(m2 + 10mn – 2) 6. (2x – 5y)(3x + y) 18s3t3 4x2y2 + 4xy3 x2 – 6x – 16 2x3 – x2 – 29x + 28 6m3n + 60m2n2 – 12mn 6x2 – 13xy – 5y2

Lesson Quiz: Part II 7. A triangle has a base that is 4cm longer than its height. a. Write a polynomial that represents the area of the triangle. 1 2 h2 + 2h b. Find the area when the height is 8 cm. 48 cm2