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Polynomials 6-5 6-5 Multiplying Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1Algebra 1 Holt McDougal 6-5 Multiplying Polynomials Warm Up Evaluate. 1. 32 2. 24 3. 102 Simplify. 4. 23 24 5. y5 y4 6. (53)2 7. (x2)4 8. –4(x – 7) Holt McDougal Algebra 1 6-5 Multiplying Polynomials Objective Multiply polynomials. Holt McDougal Algebra 1 6-5 Multiplying Polynomials Review When multiplying powers with the same base, keep the base and add the exponents. x2 x3 = x2+3 = x5 Holt McDougal Algebra 1 6-5 Multiplying Polynomials Examples Together Multiply. A. (6y3)(3y5) (6y3)(3y5) (6 3)(y3 y5) 18y8 Group factors with like bases together. Multiply. B. (3mn2) (9m2n) (3mn2)(9m2n) (3 9)(m m2)(n2 n) 27m3n3 Holt McDougal Algebra 1 Group factors with like bases together. Multiply. 6-5 Multiplying Polynomials Example 1C: Multiplying Monomials Multiply. 1 2 2 s t 4 (st) (-12 s t2) 1 2 2 2 s t t 12 t s ( ) s 4 ( 1 2 4 • −12 s • s • s ( Holt McDougal Algebra 1 ) Group factors with like bases together. )(t • t • t ) 2 2 Multiply. 6-5 Multiplying Polynomials Try these yourself!!! Multiply. a. (2r2t)(5t3) 1 2 3 2 4 5 x y 12 x z yz ( )( b. 3 Holt McDougal Algebra 1 ) 6-5 Multiplying Polynomials *To multiply a polynomial by a monomial, use the Distributive Property. Holt McDougal Algebra 1 6-5 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 McDougal Algebra 1 6-5 Multiplying Polynomials Example 2B: Multiplying a Polynomial by a Monomial Multiply. 6pq(2p – q) (6pq)(2p – q) Distribute 6pq. (6pq)2p + (6pq)(–q) (6 2)(p Group like bases together. p)(q) + (–1)(6)(p)(q q) 12p2q – 6pq2 Holt McDougal Algebra 1 Multiply. 6-5 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 xy x y 6 + 8x y 2 ( 1 2 Distribute x y . 2 ) 1 2 1 2 2 2 Group like bases x y 6 xy + x y 8 x y ( ) 2 2 together. 2 1 2 1 • 6 x • x ( y • y) + • 8 x • x2 y • y2 2 2 ( ( ) 3x3y2 + 4x4y3 Holt McDougal Algebra 1 ) ( )( Multiply. ) 6-5 Multiplying Polynomials Try Yourself!!! Multiply. a. 3ab(5a2 + b) b. 5r2s2(r – 3s) Holt McDougal Algebra 1 6-5 Multiplying Polynomials Method 1: To multiply a binomial by a binomial, Distribute more than once (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) I 3. Multiply the Inner terms. (x + 3)(x + 2) L 4. Multiply the Last terms. (x + 3)(x + 2) x 2 = 2x 3 x = 3x 3 2 = 6 (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6 F Holt McDougal Algebra 1 O I L 6-5 Multiplying Polynomials Example 3A: Multiplying Binomials Multiply. (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. Holt McDougal Algebra 1 6-5 Multiplying Polynomials Example 3B: Multiplying Binomials Multiply. (x – 4)2 (x – 4)(x – 4) Write as a product of two binomials. Use the FOIL method. (x x) + (x (–4)) + (–4 x) + (–4 (–4)) x2 – 4x – 4x + 16 Multiply. x2 – 8x + 16 Combine like terms. Holt McDougal Algebra 1 6-5 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 McDougal Algebra 1 6-5 Multiplying Polynomials Helpful Hint In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5) Holt McDougal Algebra 1 6-5 Multiplying Polynomials Try Yourself!!! Multiply. 1. (a + 3)(a – 4) 3. (x – 3)2 Holt McDougal Algebra 1 2. (2a – b2)(a + 4b2) 6-5 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 McDougal Algebra 1 6-5 Multiplying Polynomials Example Multiply. (x – 5)(x2 + 4x – 6) Holt McDougal Algebra 1 6-5 Multiplying Polynomials Example Multiply. (x + 3)3 Holt McDougal Algebra 1 6-5 Multiplying Polynomials Method 2: Box Method (2x2 + 10x – 6) and width (5x + 3): 2x2 +10x 5x +3 –6 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 Holt McDougal Algebra 1 6-5 Multiplying Polynomials Example: Box Method Multiply. (3x + 1)(x3 + 4x2 – 7) Holt McDougal Algebra 1 6-5 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. Write the formula for the A = lw area of a rectangle. A = lw A = x(x – 4) A = x2 – 4x Substitute x – 4 for l and x for w. Multiply. The area is represented by x2 – 4x. Holt McDougal Algebra 1 6-5 Multiplying Polynomials 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 A = x2 – 4x meters shorter than width . Substitute 6 for x. A = 62 – 4 6 A = 36 – 24 Simplify. A = 12 Combine terms. The area is 12 square meters. Holt McDougal Algebra 1 6-5 Multiplying Polynomials HOMEWORK PG. 427 # 26-62 (Evens), 87-89 Holt McDougal Algebra 1