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Multiplying polynomials

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This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.

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7-7 Multiplying Polynomials 7-7 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 1 Holt Algebra
7-7 Multiplying Polynomials Warm Up Evaluate. 1. 32 9 2. 24 16 3. 102 100 Simplify. 4. 23 • 24 27 5. y5 • y4 y9 6. (53)2 56 7. (x2)4 x8 8. –4(x – 7) –4x + 28 Holt Algebra 1
7-7 Multiplying Polynomials Objective Multiply polynomials. Holt Algebra 1
7-7 Multiplying Polynomials To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter. Holt Algebra 1
7-7 Multiplying Polynomials Example 1: Multiplying Monomials Multiply. A. (6y3)(3y5) (6y3)(3y5) Group factors with like bases (6 • 3)(y3 • y5) together. 18y8 Multiply. B. (3mn2) (9m2n) (3mn2)(9m2n) Group factors with like bases (3 • 9)(m • m2)(n2 • n) together. 27m3n3 Multiply. Holt Algebra 1
7-7 Multiplying Polynomials Example 1C: Multiplying Monomials Multiply. ⎛ 1 2 2 ⎞ ⎜ s t ⎟ (s t ) - 12 st 2 ( ) Group factors with like ⎝ 4 ⎠ bases together. ⎛ 1 ⎞ ⎜ 4 i - 12 ⎟ s 2 i s i s ( )(t 2 i t i t2 ) Multiply. ⎝ ⎠ Holt Algebra 1
7-7 Multiplying Polynomials Remember! When multiplying powers with the same base, keep the base and add the exponents. x2 • x3 = x2+3 = x5 Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 1 Multiply. a. (3x3)(6x2) (3x3)(6x2) Group factors with like bases together. (3 • 6)(x3 • x2) Multiply. 18x5 b. (2r2t)(5t3) Group factors with like bases (2r2t)(5t3) together. (2 • 5)(r2)(t3 • t) Multiply. 10r2t4 Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 1 Multiply. ⎛ 1 2 ⎞ 3 2 4 5 c. ⎜ x y (12 x z )( y z ⎟ ) ⎝ 3 ⎠ ⎛ 1 2 ⎞ ⎜ 3 x y⎟ 12 x 3z 2 ( )(y 4 z5 ) Group factors with ⎝ ⎠ like bases together. ⎛ 1 i ⎞ 2 i 3 ⎜ 3 12⎟ x x ⎝ ⎠ ( )(y i y )(z4 2 i z5 ) Multiply. 4 x 5y 5 z 7 Holt Algebra 1
7-7 Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property. Holt Algebra 1
7-7 Multiplying Polynomials Example 2A: Multiplying a Polynomial by a Monomial Multiply. 4(3x2 + 4x – 8) 4(3x2 + 4x – 8) Distribute 4. (4)3x2 +(4)4x – (4)8 Multiply. 12x2 + 16x – 32 Holt Algebra 1
7-7 Multiplying Polynomials Example 2B: Multiplying a Polynomial by a Monomial Multiply. 6pq(2p – q) (6pq)(2p – q) Distribute 6pq. (6pq)2p + (6pq)(–q) Group like bases together. (6 • 2)(p • p)(q) + (–1)(6)(p)(q • q) 12p2q – 6pq2 Multiply. Holt Algebra 1
7-7 Multiplying Polynomials Example 2C: Multiplying a Polynomial by a Monomial Multiply. 1 2 2 2 x y(6xy + 8 x y ) 2 1 2 2 2 1 2 x y 6xy + 8 x y ( ) Distribute x y . 2 2 ⎛ 1 2 ⎞ ⎛ 1 2 ⎞ ⎜ 2 x y⎟ (6 xy ) + ⎜ x y ⎟ 8x 2 y2 ( ) Group like bases ⎝ ⎠ ⎝ 2 ⎠ together. ⎛ 1 ⎞ 2 ⎛ 1 ⎞ ⎜ • 6 ⎟ x • x ( y • y) + ⎜ • 8⎟ x2 • x2 y • y2 ( ) ( )( ) ⎝ 2 ⎠ ⎝ 2 ⎠ 3x3y2 + 4x4y3 Multiply. Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 2 Multiply. a. 2(4x2 + x + 3) 2(4x2 + x + 3) Distribute 2. 2(4x2) + 2(x) + 2(3) Multiply. 8x2 + 2x + 6 Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 2 Multiply. b. 3ab(5a2 + b) 3ab(5a2 + b) Distribute 3ab. (3ab)(5a2) + (3ab)(b) Group like bases (3 • 5)(a • a2)(b) + (3)(a)(b • b) together. 15a3b + 3ab2 Multiply. Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 2 Multiply. c. 5r2s2(r – 3s) 5r2s2(r – 3s) Distribute 5r2s2. (5r2s2)(r) – (5r2s2)(3s) (5)(r2 • r)(s2) – (5 • 3)(r2)(s2 • s) Group like bases together. 5r3s2 – 15r2s3 Multiply. Holt Algebra 1
7-7 Multiplying Polynomials To multiply a binomial by a binomial, you can apply the Distributive Property more than once: (x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3. = x(x + 2) + 3(x + 2) Distribute x and 3 again. = x(x) + x(2) + 3(x) + 3(2) Multiply. = x2 + 2x + 3x + 6 Combine like terms. = x2 + 5x + 6 Holt Algebra 1
7-7 Multiplying Polynomials Another method for multiplying binomials is called the FOIL method. F 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 Holt Algebra 1
7-7 Multiplying Polynomials Example 3A: Multiplying Binomials Multiply. (s + 4)(s – 2) (s + 4)(s – 2) s(s – 2) + 4(s – 2) Distribute s and 4. s(s) + s(–2) + 4(s) + 4(–2) Distribute s and 4 again. s2 – 2s + 4s – 8 Multiply. s2 + 2s – 8 Combine like terms. Holt Algebra 1
7-7 Multiplying Polynomials Example 3B: Multiplying Binomials Multiply. Write as a product of (x – 4)2 two binomials. (x – 4)(x – 4) Use the FOIL method. (x • x) + (x • (–4)) + (–4 • x) + (–4 • (–4)) x2 – 4x – 4x + 8 Multiply. x2 – 8x + 8 Combine like terms. Holt Algebra 1
7-7 Multiplying Polynomials Example 3C: Multiplying Binomials Multiply. (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. Holt Algebra 1
7-7 Multiplying Polynomials Helpful Hint In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5) Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 3a Multiply. (a + 3)(a – 4) (a + 3)(a – 4) Distribute a and 3. a(a – 4)+3(a – 4) Distribute a and 3 again. a(a) + a(–4) + 3(a) + 3(–4) a2 – 4a + 3a – 12 Multiply. a2 – a – 12 Combine like terms. Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 3b Multiply. Write as a product of (x – 3)2 two binomials. (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. Holt Algebra 1
7-7 Multiplying Polynomials 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. Holt Algebra 1
7-7 Multiplying Polynomials 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 Holt Algebra 1
7-7 Multiplying Polynomials 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 Write the product of the 5x 10x3 50x2 –30x monomials in each row and +3 6x2 30x –18 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 Holt Algebra 1
7-7 Multiplying Polynomials 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. 2x2 + 10x – 6 Multiply each term in the top polynomial by 3. × 5x + 3 Multiply each term in the top 6x2 + 30x – 18 polynomial by 5x, and align + 10x3 + 50x2 – 30x like terms. 10x3 + 56x2 + 0x – 18 Combine like terms by adding vertically. 10x3 + 56x2 + – 18 Simplify. Holt Algebra 1
7-7 Multiplying Polynomials Example 4A: Multiplying Polynomials Multiply. (x – 5)(x2 + 4x – 6) (x – 5 )(x2 + 4x – 6) Distribute x and –5. x(x2 + 4x – 6) – 5(x2 + 4x – 6) Distribute x and −5 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. Holt Algebra 1
7-7 Multiplying Polynomials Example 4B: Multiplying Polynomials Multiply. (2x – 5)(–4x2 – 10x + 3) Multiply each term in the (2x – 5)(–4x2 – 10x + 3) top polynomial by –5. –4x2 – 10x + 3 Multiply each term in the x 2x – 5 top polynomial by 2x, 20x2 + 50x – 15 and align like terms. + –8x3 – 20x2 + 6x Combine like terms by –8x3 + 56x – 15 adding vertically. Holt Algebra 1
7-7 Multiplying Polynomials Example 4C: Multiplying Polynomials Multiply. (x + 3)3 [(x + 3)(x + 3)](x + 3) Write as the product of three binomials. [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. Holt Algebra 1
7-7 Multiplying Polynomials Example 4C: Multiplying Polynomials Multiply. (x + 3)3 Use the Commutative Property of (x + 3)(x2 + 6x + 9) Multiplication. x(x2 + 6x + 9) + 3(x2 + 6x + 9) Distribute the x and 3. x(x2) + x(6x) + x(9) + 3(x2) + Distribute the x and 3 3(6x) + 3(9) again. x3 + 6x2 + 9x + 3x2 + 18x + 27 Combine like terms. x3 + 9x2 + 27x + 27 Holt Algebra 1
7-7 Multiplying Polynomials Example 4D: Multiplying Polynomials Multiply. (3x + 1)(x3 – 4x2 – 7) Write the product of the x3 4x2 –7 monomials in each 3x 3x4 12x3 –21x row and column. +1 x3 4x2 –7 Add all terms inside the rectangle. 3x4 + 12x3 + x3 + 4x2 – 21x – 7 3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms. Holt Algebra 1
7-7 Multiplying Polynomials Helpful Hint 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. Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 4a Multiply. (x + 3)(x2 – 4x + 6) (x + 3 )(x2 – 4x + 6) Distribute x and 3. x(x2 – 4x + 6) + 3(x2 – 4x + 6) Distribute x and 3 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. Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 4b Multiply. (3x + 2)(x2 – 2x + 5) Multiply each term in the (3x + 2)(x2 – 2x + 5) top polynomial by 2. x2 – 2x + 5 Multiply each term in the × 3x + 2 top polynomial by 3x, 2x2 – 4x + 10 and align like terms. + 3x3 – 6x2 + 15x Combine like terms by 3x3 – 4x2 + 11x + 10 adding vertically. Holt Algebra 1
7-7 Multiplying Polynomials 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 Write the formula for the area of a rectangle. A = l•w Substitute h – 3 for w A = (h + 4)(h – 3) and h + 4 for l. A = h2 + 4h – 3h – 12 Multiply. A = h2 + h – 12 Combine like terms. The area is represented by h2 + h – 12. Holt Algebra 1
7-7 Multiplying Polynomials 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. b. Find the area of the base when the height is 5 ft. A = h2 + h – 12 Write the formula for the area A = h2 + h – 12 the base of the prism. A = 52 + 5 – 12 Substitute 5 for h. A = 25 + 5 – 12 Simplify. A = 18 Combine terms. The area is 18 square feet. Holt Algebra 1
7-7 Multiplying Polynomials 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 Write the formula for the area of a rectangle. A = l•w A = x(x – 4) Substitute x – 4 for l and x for w. A = x2 – 4x Multiply. The area is represented by x2 – 4x. Holt Algebra 1
7-7 Multiplying Polynomials Check It Out! Example 5 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 A = x2 – 4x meters shorter than width . A = 62 – 4 • 6 Substitute 6 for x. A = 36 – 24 Simplify. A = 12 Combine terms. The area is 12 square meters. Holt Algebra 1
7-7 Multiplying Polynomials Lesson Quiz: Part I Multiply. 1. (6s2t2)(3st) 18s3t3 2. 4xy2(x + y) 4x2y2 + 4xy3 3. (x + 2)(x – 8) x2 – 6x – 16 4. (2x – 7)(x2 + 3x – 4) 2x3 – x2 – 29x + 28 5. 6mn(m2 + 10mn – 2) 6m3n + 60m2n2 – 12mn 6. (2x – 5y)(3x + y) 6x2 – 13xy – 5y2 Holt Algebra 1
7-7 Multiplying Polynomials 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 h + 2h 2 b. Find the area when the height is 8 cm. 48 cm2 Holt Algebra 1

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Multiplying polynomials

  • 1. 7-7 Multiplying Polynomials 7-7 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 1 Holt Algebra
  • 2. 7-7 Multiplying Polynomials Warm Up Evaluate. 1. 32 9 2. 24 16 3. 102 100 Simplify. 4. 23 • 24 27 5. y5 • y4 y9 6. (53)2 56 7. (x2)4 x8 8. –4(x – 7) –4x + 28 Holt Algebra 1
  • 3. 7-7 Multiplying Polynomials Objective Multiply polynomials. Holt Algebra 1
  • 4. 7-7 Multiplying Polynomials To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter. Holt Algebra 1
  • 5. 7-7 Multiplying Polynomials Example 1: Multiplying Monomials Multiply. A. (6y3)(3y5) (6y3)(3y5) Group factors with like bases (6 • 3)(y3 • y5) together. 18y8 Multiply. B. (3mn2) (9m2n) (3mn2)(9m2n) Group factors with like bases (3 • 9)(m • m2)(n2 • n) together. 27m3n3 Multiply. Holt Algebra 1
  • 6. 7-7 Multiplying Polynomials Example 1C: Multiplying Monomials Multiply. ⎛ 1 2 2 ⎞ ⎜ s t ⎟ (s t ) - 12 st 2 ( ) Group factors with like ⎝ 4 ⎠ bases together. ⎛ 1 ⎞ ⎜ 4 i - 12 ⎟ s 2 i s i s ( )(t 2 i t i t2 ) Multiply. ⎝ ⎠ Holt Algebra 1
  • 7. 7-7 Multiplying Polynomials Remember! When multiplying powers with the same base, keep the base and add the exponents. x2 • x3 = x2+3 = x5 Holt Algebra 1
  • 8. 7-7 Multiplying Polynomials Check It Out! Example 1 Multiply. a. (3x3)(6x2) (3x3)(6x2) Group factors with like bases together. (3 • 6)(x3 • x2) Multiply. 18x5 b. (2r2t)(5t3) Group factors with like bases (2r2t)(5t3) together. (2 • 5)(r2)(t3 • t) Multiply. 10r2t4 Holt Algebra 1
  • 9. 7-7 Multiplying Polynomials Check It Out! Example 1 Multiply. ⎛ 1 2 ⎞ 3 2 4 5 c. ⎜ x y (12 x z )( y z ⎟ ) ⎝ 3 ⎠ ⎛ 1 2 ⎞ ⎜ 3 x y⎟ 12 x 3z 2 ( )(y 4 z5 ) Group factors with ⎝ ⎠ like bases together. ⎛ 1 i ⎞ 2 i 3 ⎜ 3 12⎟ x x ⎝ ⎠ ( )(y i y )(z4 2 i z5 ) Multiply. 4 x 5y 5 z 7 Holt Algebra 1
  • 10. 7-7 Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property. Holt Algebra 1
  • 11. 7-7 Multiplying Polynomials Example 2A: Multiplying a Polynomial by a Monomial Multiply. 4(3x2 + 4x – 8) 4(3x2 + 4x – 8) Distribute 4. (4)3x2 +(4)4x – (4)8 Multiply. 12x2 + 16x – 32 Holt Algebra 1
  • 12. 7-7 Multiplying Polynomials Example 2B: Multiplying a Polynomial by a Monomial Multiply. 6pq(2p – q) (6pq)(2p – q) Distribute 6pq. (6pq)2p + (6pq)(–q) Group like bases together. (6 • 2)(p • p)(q) + (–1)(6)(p)(q • q) 12p2q – 6pq2 Multiply. Holt Algebra 1
  • 13. 7-7 Multiplying Polynomials Example 2C: Multiplying a Polynomial by a Monomial Multiply. 1 2 2 2 x y(6xy + 8 x y ) 2 1 2 2 2 1 2 x y 6xy + 8 x y ( ) Distribute x y . 2 2 ⎛ 1 2 ⎞ ⎛ 1 2 ⎞ ⎜ 2 x y⎟ (6 xy ) + ⎜ x y ⎟ 8x 2 y2 ( ) Group like bases ⎝ ⎠ ⎝ 2 ⎠ together. ⎛ 1 ⎞ 2 ⎛ 1 ⎞ ⎜ • 6 ⎟ x • x ( y • y) + ⎜ • 8⎟ x2 • x2 y • y2 ( ) ( )( ) ⎝ 2 ⎠ ⎝ 2 ⎠ 3x3y2 + 4x4y3 Multiply. Holt Algebra 1
  • 14. 7-7 Multiplying Polynomials Check It Out! Example 2 Multiply. a. 2(4x2 + x + 3) 2(4x2 + x + 3) Distribute 2. 2(4x2) + 2(x) + 2(3) Multiply. 8x2 + 2x + 6 Holt Algebra 1
  • 15. 7-7 Multiplying Polynomials Check It Out! Example 2 Multiply. b. 3ab(5a2 + b) 3ab(5a2 + b) Distribute 3ab. (3ab)(5a2) + (3ab)(b) Group like bases (3 • 5)(a • a2)(b) + (3)(a)(b • b) together. 15a3b + 3ab2 Multiply. Holt Algebra 1
  • 16. 7-7 Multiplying Polynomials Check It Out! Example 2 Multiply. c. 5r2s2(r – 3s) 5r2s2(r – 3s) Distribute 5r2s2. (5r2s2)(r) – (5r2s2)(3s) (5)(r2 • r)(s2) – (5 • 3)(r2)(s2 • s) Group like bases together. 5r3s2 – 15r2s3 Multiply. Holt Algebra 1
  • 17. 7-7 Multiplying Polynomials To multiply a binomial by a binomial, you can apply the Distributive Property more than once: (x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3. = x(x + 2) + 3(x + 2) Distribute x and 3 again. = x(x) + x(2) + 3(x) + 3(2) Multiply. = x2 + 2x + 3x + 6 Combine like terms. = x2 + 5x + 6 Holt Algebra 1
  • 18. 7-7 Multiplying Polynomials Another method for multiplying binomials is called the FOIL method. F 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 Holt Algebra 1
  • 19. 7-7 Multiplying Polynomials Example 3A: Multiplying Binomials Multiply. (s + 4)(s – 2) (s + 4)(s – 2) s(s – 2) + 4(s – 2) Distribute s and 4. s(s) + s(–2) + 4(s) + 4(–2) Distribute s and 4 again. s2 – 2s + 4s – 8 Multiply. s2 + 2s – 8 Combine like terms. Holt Algebra 1
  • 20. 7-7 Multiplying Polynomials Example 3B: Multiplying Binomials Multiply. Write as a product of (x – 4)2 two binomials. (x – 4)(x – 4) Use the FOIL method. (x • x) + (x • (–4)) + (–4 • x) + (–4 • (–4)) x2 – 4x – 4x + 8 Multiply. x2 – 8x + 8 Combine like terms. Holt Algebra 1
  • 21. 7-7 Multiplying Polynomials Example 3C: Multiplying Binomials Multiply. (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. Holt Algebra 1
  • 22. 7-7 Multiplying Polynomials Helpful Hint In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5) Holt Algebra 1
  • 23. 7-7 Multiplying Polynomials Check It Out! Example 3a Multiply. (a + 3)(a – 4) (a + 3)(a – 4) Distribute a and 3. a(a – 4)+3(a – 4) Distribute a and 3 again. a(a) + a(–4) + 3(a) + 3(–4) a2 – 4a + 3a – 12 Multiply. a2 – a – 12 Combine like terms. Holt Algebra 1
  • 24. 7-7 Multiplying Polynomials Check It Out! Example 3b Multiply. Write as a product of (x – 3)2 two binomials. (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. Holt Algebra 1
  • 25. 7-7 Multiplying Polynomials 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. Holt Algebra 1
  • 26. 7-7 Multiplying Polynomials 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 Holt Algebra 1
  • 27. 7-7 Multiplying Polynomials 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 Write the product of the 5x 10x3 50x2 –30x monomials in each row and +3 6x2 30x –18 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 Holt Algebra 1
  • 28. 7-7 Multiplying Polynomials 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. 2x2 + 10x – 6 Multiply each term in the top polynomial by 3. × 5x + 3 Multiply each term in the top 6x2 + 30x – 18 polynomial by 5x, and align + 10x3 + 50x2 – 30x like terms. 10x3 + 56x2 + 0x – 18 Combine like terms by adding vertically. 10x3 + 56x2 + – 18 Simplify. Holt Algebra 1
  • 29. 7-7 Multiplying Polynomials Example 4A: Multiplying Polynomials Multiply. (x – 5)(x2 + 4x – 6) (x – 5 )(x2 + 4x – 6) Distribute x and –5. x(x2 + 4x – 6) – 5(x2 + 4x – 6) Distribute x and −5 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. Holt Algebra 1
  • 30. 7-7 Multiplying Polynomials Example 4B: Multiplying Polynomials Multiply. (2x – 5)(–4x2 – 10x + 3) Multiply each term in the (2x – 5)(–4x2 – 10x + 3) top polynomial by –5. –4x2 – 10x + 3 Multiply each term in the x 2x – 5 top polynomial by 2x, 20x2 + 50x – 15 and align like terms. + –8x3 – 20x2 + 6x Combine like terms by –8x3 + 56x – 15 adding vertically. Holt Algebra 1
  • 31. 7-7 Multiplying Polynomials Example 4C: Multiplying Polynomials Multiply. (x + 3)3 [(x + 3)(x + 3)](x + 3) Write as the product of three binomials. [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. Holt Algebra 1
  • 32. 7-7 Multiplying Polynomials Example 4C: Multiplying Polynomials Multiply. (x + 3)3 Use the Commutative Property of (x + 3)(x2 + 6x + 9) Multiplication. x(x2 + 6x + 9) + 3(x2 + 6x + 9) Distribute the x and 3. x(x2) + x(6x) + x(9) + 3(x2) + Distribute the x and 3 3(6x) + 3(9) again. x3 + 6x2 + 9x + 3x2 + 18x + 27 Combine like terms. x3 + 9x2 + 27x + 27 Holt Algebra 1
  • 33. 7-7 Multiplying Polynomials Example 4D: Multiplying Polynomials Multiply. (3x + 1)(x3 – 4x2 – 7) Write the product of the x3 4x2 –7 monomials in each 3x 3x4 12x3 –21x row and column. +1 x3 4x2 –7 Add all terms inside the rectangle. 3x4 + 12x3 + x3 + 4x2 – 21x – 7 3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms. Holt Algebra 1
  • 34. 7-7 Multiplying Polynomials Helpful Hint 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. Holt Algebra 1
  • 35. 7-7 Multiplying Polynomials Check It Out! Example 4a Multiply. (x + 3)(x2 – 4x + 6) (x + 3 )(x2 – 4x + 6) Distribute x and 3. x(x2 – 4x + 6) + 3(x2 – 4x + 6) Distribute x and 3 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. Holt Algebra 1
  • 36. 7-7 Multiplying Polynomials Check It Out! Example 4b Multiply. (3x + 2)(x2 – 2x + 5) Multiply each term in the (3x + 2)(x2 – 2x + 5) top polynomial by 2. x2 – 2x + 5 Multiply each term in the × 3x + 2 top polynomial by 3x, 2x2 – 4x + 10 and align like terms. + 3x3 – 6x2 + 15x Combine like terms by 3x3 – 4x2 + 11x + 10 adding vertically. Holt Algebra 1
  • 37. 7-7 Multiplying Polynomials 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 Write the formula for the area of a rectangle. A = l•w Substitute h – 3 for w A = (h + 4)(h – 3) and h + 4 for l. A = h2 + 4h – 3h – 12 Multiply. A = h2 + h – 12 Combine like terms. The area is represented by h2 + h – 12. Holt Algebra 1
  • 38. 7-7 Multiplying Polynomials 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. b. Find the area of the base when the height is 5 ft. A = h2 + h – 12 Write the formula for the area A = h2 + h – 12 the base of the prism. A = 52 + 5 – 12 Substitute 5 for h. A = 25 + 5 – 12 Simplify. A = 18 Combine terms. The area is 18 square feet. Holt Algebra 1
  • 39. 7-7 Multiplying Polynomials 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 Write the formula for the area of a rectangle. A = l•w A = x(x – 4) Substitute x – 4 for l and x for w. A = x2 – 4x Multiply. The area is represented by x2 – 4x. Holt Algebra 1
  • 40. 7-7 Multiplying Polynomials Check It Out! Example 5 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 A = x2 – 4x meters shorter than width . A = 62 – 4 • 6 Substitute 6 for x. A = 36 – 24 Simplify. A = 12 Combine terms. The area is 12 square meters. Holt Algebra 1
  • 41. 7-7 Multiplying Polynomials Lesson Quiz: Part I Multiply. 1. (6s2t2)(3st) 18s3t3 2. 4xy2(x + y) 4x2y2 + 4xy3 3. (x + 2)(x – 8) x2 – 6x – 16 4. (2x – 7)(x2 + 3x – 4) 2x3 – x2 – 29x + 28 5. 6mn(m2 + 10mn – 2) 6m3n + 60m2n2 – 12mn 6. (2x – 5y)(3x + y) 6x2 – 13xy – 5y2 Holt Algebra 1
  • 42. 7-7 Multiplying Polynomials 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 h + 2h 2 b. Find the area when the height is 8 cm. 48 cm2 Holt Algebra 1