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: 89



Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $ \dfrac{x\cdot2x-(x^2+1)\cdot1}{(x^2+1)^2} ,$ multiply the factors first. Then remove the grouping symbols and combine like terms. Finally, use special products to simplify the denominator. $\bf{\text{Solution Details:}}$ Multiplying the factors, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{2x^2-(x^2+1)}{(x^2+1)^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-x^2-1}{(x^2+1)^2} \\\\= \dfrac{x^2-1}{(x^2+1)^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{x^2-1}{(x^2)^2+2(x^2)(1)+(1)^2} \\\\= \dfrac{x^2-1}{x^4+2x^2+1} .\end{array}
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