Q. 41 - Precalculus Enhanced with Graphing Utilities

Question

Write the partial fraction decomposition of each rational expression. 

$\frac{x^3}{{(x2+16)}^3}$

Step-by-Step Solution

Answer

The partial fraction decomposition is  $\frac{x^3}{{(x2+16)}^3}=\frac{x}{{(x^2+16)}^2}+\frac{-16x}{{(x^2+16)}^3}$

Step 1. Given Information.

The rational expression is,  

$\frac{x^3}{{(x^2+16)}^3}=\frac{Ax+B}{x^2+16}+\frac{Cx+D}{{(x^2+16)}^2}+\frac{Ex+F}{{(x^2+16)}^3}$

$\frac{x^3}{{(x^2+16)}^3}=\frac{{(x^2+16)}^2}{(Ax+B)}+\frac{(x^2+16)}{(Cx+D)}+\frac{Ex+F}{{(x^2+16)}^3}$

Therefore,

$x^3={(x^2+16)}^2(Ax+B)+(x^2+16)(Cx+D)+Ex+F$

$x^3=x^{5}A+x^{4}B+32x^{3}A+x^{3}C+32x^{2}B+x^{2}D+256xA+16xC+xE+256B+16D+F$

Therefore,

$$\begin{gathered} A=0 \\ B=0 \\ 32A+C=132 \\ B+D=0 \\ 256A+16C+E=0 \\ 256B+16D+F=0 \end{gathered}$$

Step 2. Partial Fraction Decomposition.

Decompose the rational fraction, we get,  

$A=0 , B=0, C=1, D=0, E=-16, F=0$

Therefore,

$\frac{x^3}{{(x2+16)}^3}=\frac{x}{{(x^2+16)}^2}+\frac{-16x}{{(x^2+16)}^3}$