Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 11 - Systems of Equations and Inequalities - 11.5 Partial Fraction Decomposition - 11.5 Assess Your Understanding - Page 768: 8


Improper; Proper Form: $3+\frac{1}{x^2-1}$

Work Step by Step

A rational expression is proper if and only if the degree of its numerator's polynomial is less than the degree of its denominator's polynomial. Here, the degree of the numerator is $2$ while the degree of the denominator is also $2$ Hence, the rational expression is improper. Make the rational expression proper by rewriting then simplify: \begin{align*} \frac{3x^2-2}{x^2-1}&=\frac{3x^2-3+1}{x^2-1}\\ &=\dfrac{3(x^2-1)+1}{x^2-1}\\ &=\frac{3(x^2-1)}{x^2-1}+\frac{1}{x^2-1}\\ &=3+\frac{1}{x^2-1} \end{align*}
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