College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter R - Section R.7 - Rational Expressions - R.7 Assess Your Understanding - Page 72: 94



Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $ \dfrac{(x^2+9)\cdot2-(2x-5)\cdot2x}{(x^2+9)^2} ,$ use the Distributive Property first. Then remove the grouping symbols and combine like terms. Finally, use special products to simplify the denominator. $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{(x^2\cdot2+9\cdot2)-(2x\cdot2x-5\cdot2x)}{(x^2+9)^2} \\\\= \dfrac{(2x^2+18)-(4x^2-10x)}{(x^2+9)^2} .\end{array} Removing the grouping symbols and then combining like terms, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{2x^2+18-4x^2+10x}{(x^2+9)^2} \\\\= \dfrac{-2x^2+10x+18}{(x^2+9)^2} .\end{array} Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{-2x^2+10x+18}{(x^2)^2+2(x^2)(9)+(9)^2} \\\\= \dfrac{-2x^2+10x+18}{x^4+18x^2+81} .\end{array}
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