Q. 53 - Precalculus Enhanced with Graphing Utilities

Question

In Problems, use the division algorithm to rewrite each improper fraction as the sum of a quotient and proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition.

$\frac{x^5+x^4-x^2+2}{x^4-2x^2+1}$

Step-by-Step Solution

Answer

Expression $\frac{x^5+x^4-x^2+2}{x^4-2x^2+1}$as the sum of a quotient and the partial fraction decomposition is $\frac{x^5+x^4-x^2+2}{x^4-2x^2+1}=x+1+\frac{1.25}{(x-1)}+\frac{0.25}{{(x-1)}^2}+\frac{0.75}{(x+1)}+\frac{0.25}{{(x+1)}^2}$

Step 1. Given data

The given expression is 

$\frac{x^5+x^4-x^2+2}{x^4-2x^2+1}$

Step 2. Division

Division Algorithm 

$\frac{x^5+x^4-x^2+2}{x^4-2x^2+1}=x+1+\frac{2x^3+x^2-x+1}{x^4-2x^2+1}$

Step 3. Partial fraction

partial fraction decomposition of the proper fraction 

$$\begin{gathered} \frac{2x^3+x^2-x+1}{x^4-2x^2+1}=\frac{2x^3+x^2-x+1}{{(x-1)}^2{(x+1)}^2} \\ \frac{2x^3+x^2-x+1}{{(x-1)}^2{(x+1)}^2}=\frac{A}{(x-1)}+\frac{B}{{(x-1)}^2}+\frac{C}{(x+1)}+\frac{D}{{(x+1)}^2} \\ 2x^3+x^2-x+1=A(x-1){(x+1)}^2+B{(x+1)}^2+C(x+1){(x-1)}^2+D{(x-1)}^2 \end{gathered}$$

Solving A,B,C, and D

$A=1.25, B=0.25, C=0.75, \& D=0.25$

So $\frac{x^5+x^4-x^2+2}{x^4-2x^2+1}=x+1+\frac{1.25}{(x-1)}+\frac{0.25}{{(x-1)}^2}+\frac{0.75}{(x+1)}+\frac{0.25}{{(x+1)}^2}$