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Chapter 5 Section 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.5 Multiplying Polynomials 1 Multiply a monomial and a polynomial. 2 Multiply two polynomials. 3 Multiply binomials by the FOIL method. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Multiply a monomial and a polynomial. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 3 Multiply a monomial and a polynomial. As shown in Section 5.1, we find the product of two monomials by using the rules for exponents and the commutative and associative properties. For example 8m6 9n6 8 9 m6 n 6 72m6 n 6 . To find the product of a monomial and a polynomial with more than one term we use the distributive property and multiplication of monomials. Do not confuse addition of terms with multiplication of terms. For instance, 7q5 2q5 9q5 , but 7q 2q 7 2q 5 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 55 14q10 . Slide 5.5 - 4 EXAMPLE 1 Multiplying Monomials and Polynomials Find the product. 2 x 4 3x 2 2 x 5 Solution: 2 x 3x 2 x 2 x 2 x 5 4 2 6 x 4 x 10 x 6 5 4 4 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 5 Objective 2 Multiply two polynomials. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 6 Multiply two polynomials. We can use the distributive property repeatedly to find the product of any two polynomials. For example, to find the product of the polynomials x2 + 3x +5 and x − 4, think of x − 4 as a single quantity and use the distributive property as follows. x 2 3x 5 x 4 x 2 x 4 3x x 4 5 x 4 Now use the distributive property three more times to find x2(x − 4), 3x(x − 4), and 5(x − 4). x2 x x2 4 3x x 3x 4 5 x 5 4 x3 4 x 2 3x 2 12 x 5 x 20 x3 x2 7 x 20 This example suggests the following rule. To multiply polynomials, multiply each term of the second polynomial by each term of the first polynomial and add the products. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 7 EXAMPLE 2 Multiplying Two Polynomials Multiply (m3 − 2m + 1)·(2m2 + 4m + 3). Solution: m3 2m2 m3 4m m3 3 2m 2m2 2m 4m 2m 3 1 2m2 1 4m 1 3 2m5 4m 4 3m3 4m3 8m 2 6m 2m 2 4m 3 2m5 4m4 m3 6m2 2m 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 8 EXAMPLE 3 Multiplying Polynomials Vertically Multiply. 3x 2 4 x 5 x4 Solution: 12 x2 16 x 20 3x3 4 x 2 5 x 3x3 16 x2 11x 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 9 EXAMPLE 4 Multiplying Polynomials with Fractional Coefficients Vertically Multiply. 5x3 10 x 2 20 1 2 2 x 5 5 Solution: 2 x3 4 x 2 8 x5 2 x 4 0 x3 4 x 2 x5 2 x 4 2 x3 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 10 EXAMPLE 4A Multiplying Polynomials with a Rectangle Model Use the rectangle method to find the product 4x 3 x 2. Solution: 4x 2 3x 8x 6 4 x2 11x 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 11 Objective 3 Multiply binomials by the FOIL method. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 12 Multiply binomials by the FOIL method. In algebra, many times the polynomials to be multiplied are binomials. For these products, the FOIL method reduces the rectangle method to a systematic approach without the rectangle. A summary of the steps in the FOIL method follows. Step 1: Multiply the two First terms of the binomials to get the first term of the answer. Step 2: Find the Outer product and Inner product and add them (when possible) to get the middle term of the answer. Step 3: Multiply the two Last terms of the binomials to get the last term of the answer. L 15 F x2 x 3 x 5 O 5x I 3x Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 13 EXAMPLE 5 Using the FOIL Method Find the product by the FOIL method. F x L 12 2 x 2 x 6 O 6x I 2x Solution: x2 6 x 2 x 8 x 8x 12 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 14 EXAMPLE 6 Using the FOIL Method Multiply 5x 6 2 y 3 . F 10xy L 18 5x 6 2 y 3 O 15x I 12 y Solution: 10 xy 15x 12 y 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 15 EXAMPLE 7 Using the FOIL Method Find each product. 4 y x 2 y 3x Solution: 8 y 12xy 2xy 3x 8 y 2 14xy 3x2 3 3x x 2 2x 1 2 2 3 x 3 2 x 2 1x 4 x 2 3x3 2 x 2 3x 2 6x 9x 6x 5 4 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.5 - 16