Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.4 - Rational Expressions - 1.4 Exercises - Page 43: 24



Work Step by Step

$\dfrac{1-x^{2}}{x^{3}-1}$ We factor the numerator using the difference of squares factoring formula, which is: $a^{2}-b^{2}=(a-b)(a+b)$. Also, we factor the denominator using the difference of cubes factoring formula, which is: $a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$ $\dfrac{1-x^{2}}{x^{3}-1}=\dfrac{(1-x)(1+x)}{(x-1)(x^{2}+x+1)}=...$ We are allowed to change the sign of the numerator or denominator if we also change the sign of the fraction. Knowing that, we rewrite the expression and simplify: $...=-\dfrac{-(1-x)(1+x)}{(x-1)(x^{2}+x+1)}=-\dfrac{(x-1)(1+x)}{(x-1)(x^{2}+x+1)}=...$ $...=-\dfrac{x+1}{x^{2}+x+1}$
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