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

Warm Up Evaluate. 1. 32 3. 102 Simplify. 4. 23  24 6. (53)2 9 2. 24 16 100 27 5. y5  y4 y9 56 7. (x2)4 x8 8. –4(x – 7) –4x + 28

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.

Example 1C: Multiplying Monomials 4 s2 t2 (st) (-12 s t2) ( ) æ ç è - 2 1 12 4 t s ö ÷ ø Group factors with like bases together. ( ) • æ ö ç è 2 1 −12 4 t s ÷ ø • • Multiply.

When multiplying powers with the same base, keep the base and add the exponents. x2  x3 = x2+3 = x5 Remember!

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

Example 2B: Multiplying a Polynomial by a Monomial 6pq(2p – q) Distribute 6pq. (6pq)(2p – q) (6pq)2p + (6pq)(–q) Group like bases together. (6  2)(p  p)(q) + (–1)(6)(p)(q  q) 12p2q – 6pq2 Multiply.

Example 2C: Multiplying a Polynomial by a Monomial 1 ( ) x y 2 6 xy + 8 x y 2 2 2 Distribute . 2 1 x y x y ( ) + 2 6 1 xy y x 8 x y x y ( ) æ ç è + 2 1 6 8 xy ö ÷ ø Group like bases together. x2 • x ( ) æ + ç è 1 • 6 2 y • y x2 • x2 y • y2 • 8 ö ÷ ø 3x3y2 + 4x4y3 Multiply.

Check It Out! Example 2 Multiply. a. 2(4x2 + x + 3) Distribute 2. 2(4x2 + x + 3) 2(4x2) + 2(x) + 2(3) Multiply. 8x2 + 2x + 6

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.

Check It Out! Example 2 Multiply. c. 5r2s2(r – 3s) Distribute 5r2s2. 5r2s2(r – 3s) (5r2s2)(r) – (5r2s2)(3s) Group like bases together. (5)(r2  r)(s2) – (5  3)(r2)(s2  s) 5r3s2 – 15r2s3 Multiply.

To multiply a binomial by a binomial, you can apply the Distributive Property more than once: Distribute. (x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute again. = x(x + 2) + 3(x + 2) = x(x) + x(2) + 3(x) + 3(2) Multiply. = x2 + 2x + 3x + 6 Combine like terms. = x2 + 5x + 6

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

***Another method: Using table***

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

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