Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 7 - Section 7.3 - Partial Fractions - Exercise Set - Page 841: 7


The partial fraction expansion is $\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}=\frac{Ax+B}{{{x}^{2}}+4}+\frac{Cx+D}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$.

Work Step by Step

The provided rational expression is as given below: $\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$ Now, solve the expression as follows: We set up the partial fraction expansion with unknown constants coefficients and then write a constant coefficients over each of the two distinct algebraic linear factors in the denominator of the expression. Then, decompose the fractional part as follows: $\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}=\frac{Ax+B}{{{x}^{2}}+4}+\frac{Cx+D}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$ Thus, $\frac{Ax+B}{{{x}^{2}}+4}+\frac{Cx+D}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$ is a partial fraction expansion of rational expression $\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$ with constants $A$,$B$,$C$ and $D$.
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