Q. 50 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 47–54, 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 

$\frac{x^3+x}{x^2+4}$

Step-by-Step Solution

Answer

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

Step 1: Given information

Improper fraction $\frac{x^3+x}{x^2+4}$

Step 2: Express in partial fraction decomposition.

First, we have to convert the improper fraction as the sum of a quotient and a proper fraction. 

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

Now, find the partial fraction of the second term.

$$\begin{gathered} \frac{3x}{x^2+4}=\frac{Ax+B}{x^2+4} \\ 3x=Ax+B \end{gathered}$$

After comparing the coefficient of variable and constant.

We get that,

$A=3, B=0$

So that, 

$\frac{3x}{x^2+4}=\frac{3x+0}{x^2+4}=\frac{3x}{x^2+4}$

So that, given improper fraction can be written as,

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