Exponents and Polynomials - XYZ Custom Plus

Multiplying Polynomials: An IntroductionA Multiplying Polynomials AlgebraicallyRecall that the distributive property allows us to multiply across parentheseswhen a sum or difference is enclosed within the parentheses. That is:a(b + c) = a ⋅ b + a ⋅ cWe can use the distributive property to multiply polynomials.Example 1Multiply: x 2 (x 3 + x 4 )9.3ObjectivesA Multiply polynomials algebraically.B Multiply binomials.C Multiply polynomials geometrically.Examples now playing atMathTV.com/booksSolutionApplying the distributive property, we havex 2 (x 3 + x 4 ) = x 2 ⋅ x 3 + x 2 ⋅ x 4 Distributive property= x 5 + x 6Practice Problems1. Multiply: x 3 (x 5 + x 7 )The distributive property works for multiplication from the right as well as theleft. That is, we can also write the distributive property this way:(b + c)a = b ⋅ a + c ⋅ aExample 2Multiply: (x 3 + x 4 )x 2Solution Because multiplication is a commutative operation, we shouldexpect to obtain the same answer as in Example 1 above.2. Multiply: (x 5 + x 7 )x 3(x 3 + x 4 )x 2 = x 3 ⋅ x 2 + x 4 ⋅ x 2= x 5 + x 6Example 3Multiply: 4x 3 (6x 2 − 8)3. Multiply: 5x 2 (6x 3 − 4)Solution The distributive property allows us to multiply 4x 3 by both 6x 2and 8:4x 3 (6x 2 − 8) = 4x 3 ⋅ 6x 2 − 4x 3 ⋅ 8= (4 ⋅ 6)(x 3 ⋅ x 2 ) − (4 ⋅ 8)x 3= 24x 5 − 32x 3Example 4Multiply: 2a 4 b 2 (3a 3 + 4b 5 )Solution Applying the distributive property as we did in the previous twoexamples, we have4. Multiply: 5a 3 b 5 (2a 2 + 7b 2 )2a 4 b 2 (3a 3 + 4b 5 ) = 2a 4 b 2 ⋅ 3a 3 + 2a 4 b 2 ⋅ 4b 5= (2 ⋅ 3)(a 4 ⋅ a 3 )b 2 + (2 ⋅ 4)(a 4 )(b 2 ⋅ b 5 )= 6a 7 b 2 + 8a 4 b 7B Multiplying BinomialsPolynomials with exactly two terms are called binomials. We multiply binomialsby applying the distributive property.Answers1. x 8 + x 10 2. x 8 + x 103. 30x 5 − 20x 24. 10a 5 b 5 + 35a 3 b 79.3 Multiplying Polynomials: An Introduction581