Chapter 11.5, Problem 50AYU

In Problems 47-54, use the division algorithm to rewrite each improper rational expression as the sum of a polynomial and a proper rational expression. Find the partial fraction decomposition of the proper rational expression. Finally, express the improper rational expression as the sum of a polynomial and the partial fraction decomposition. $\frac{x^3+x}{x^2+4}$

To determine

To find:

i. Rewrite improper rational expression as the sum of polynomial and a proper rational expression using the division algorithm.

ii. The partial fraction decomposition of the proper rational expression

iii. To express the improper rational expression as the sum of polynomial and the partial fraction decomposition.

Answer to Problem 50AYU

Solution:

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

Explanation of Solution

Given:

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

Calculation:

Divisional Algorithm

From the above division,

$\frac{x^3 + x}{x^2 + 4} = x - \frac{3x}{x^2 + 4}$   -----Eq (1)

$\frac{3x}{x^2 + 4}$ is already in decomposed form. Hence it cannot be further decomposed into partial fractions.

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