74 CHAPTER R Review of Basic Concepts Multiplication and Division We multiply and divide rational expressions using the same properties used for multiplying and dividing fractions. Multiplying and Dividing Rational Expressions For fractions a b and c d 1b≠0, d≠02, the following hold true. a b # c d = ac bd and a b ÷ c d = a b # d c 1 c 302 That is, to find the product of two fractions, multiply their numerators to find the numerator of the product. Then multiply their denominators to find the denominator of the product. To divide two fractions, multiply the dividend (the first fraction) by the reciprocal of the divisor (the second fraction). EXAMPLE 3 Multiplying or Dividing Rational Expressions Multiply or divide, as indicated. (a) 2y2 9 # 27 8y5 (b) 3m2 - 2m- 8 3m2 + 14m+ 8 # 3m+ 2 3m+ 4 (c) 3p2 + 11p - 4 24p3 - 8p2 , 9p + 36 24p4 - 36p3 (d) x3 - y3 x2 - y2 # 2x + 2y + xz + yz 2x2 + 2y2 + zx2 + zy2 SOLUTION (a) 2y2 9 # 27 8y5 = 2y2 # 27 9 # 8y5 Multiply fractions. = 2 # 9 # 3 # y2 9 # 2 # 4 # y2 # y3 Factor. = 3 4y3 Lowest terms As shown here, it is generally easier to divide out any common factors in the numerator and denominator before performing the actual multiplication. (b) 3m2 - 2m- 8 3m2 + 14m+ 8 # 3m+ 2 3m+ 4 = 1m- 2213m+ 42 1m+ 4213m+ 22 # 3m+ 2 3m+ 4 Factor. = 1m- 2213m+ 4213m+ 22 1m+ 4213m+ 2213m+ 42 Multiply fractions. = m- 2 m+ 4 Lowest terms
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