Q. 49 - Precalculus Enhanced with Graphing Utilities

Question

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.

$\frac{x^3}{x^2+1}$

Step-by-Step Solution

Answer

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

Step 1. Given information.

Improper fraction $\frac{x^3}{x^2+1}$

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. 

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

Now, find the partial fraction of the second term.

$$\begin{gathered} \frac{x}{x^2+1}=\frac{Ax+B}{x^2+1} \\ x=Ax+B \end{gathered}$$

After comparing the coefficient of variable and constant.

We get that,

       $A=1, B=0$

Therefore,

$\frac{x}{x^2+1}=\frac{x}{x^2+1}$

So that, given improper fraction can be written as,

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