College Algebra (10th Edition)

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

Chapter 8 - Section 8.5 - Partial Fraction Decomposition - 8.5 Assess Your Understanding - Page 608: 9


$ \displaystyle \frac{5x^{3}+2x-1}{x^{2}-4}$ is improper$, $ $\displaystyle \frac{5x^{3}+2x-1}{x^{2}-4}=5x+\frac{22x-1}{x^{2}-4}$

Work Step by Step

The rational expression $\displaystyle \frac{P}{Q}$ is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. Otherwise, the rational expression is called improper. --- The degree of the numerator, $P(x)=5x^{3}+2x-1, \quad $is $3$, and the degree of the denominator$ , Q(x)=x^{2}-4, \quad $is $2$. The rational expression is improper. To make it proper, $\displaystyle \frac{5x^{3}+2x-1}{x^{2}-4} = \frac{5x(x^{2}-4)+20x+2x-1}{x^{2}-4}=\\\\\\ =\dfrac{5x(x^{2}-4)}{x^{2}-4}+\dfrac{22x-1}{x^{2}-4}$ $=5x+\displaystyle \frac{22x-1}{x^{2}-4}$
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