## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 7 - Section 7.3 - Partial Fractions - Concept and Vocabulary Check - Page 841: 2

#### Answer

The stated partial fraction decomposition is incorrect.

#### Work Step by Step

It is provided that $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{\left( x-4 \right)}$ is the partial fraction decomposition of the algebraic expression $\frac{3x}{\left( x+5 \right){{\left( x-4 \right)}^{2}}}$ into a series of smaller components. We know that the partial fraction decomposition is the technique used for rational functions. It is the mathematical operation by which rational function can be expressed as a sum of a polynomial and several fractions, including ‘equating coefficients’. And the partial fraction decomposition associated with $\frac{p{{x}^{2}}+qx+r}{\left( x-a \right){{\left( x-b \right)}^{2}}},a\ne b$, is: $\frac{A}{\left( x-a \right)}+\frac{B}{\left( x-b \right)}+\frac{C}{{{\left( x-b \right)}^{2}}}$. Therefore, $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{{{\left( x-4 \right)}^{2}}}$ is the final simplified algebraic expression Thus, the stated partial fraction decomposition $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{\left( x-4 \right)}$ of $\frac{3x}{\left( x+5 \right){{\left( x-4 \right)}^{2}}}$ is incorrect and the correct partial fraction decomposition is as follows: $\frac{3x}{\left( x+5 \right){{\left( x-4 \right)}^{2}}}$ $=$ $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{{{\left( x-4 \right)}^{2}}}$ .

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