Chapter 11.5, Problem 12AYU

In Problems 5-12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression. $\frac{2x\left(x^2+4\right)}{x^2+1}$

To determine

To find: The given rational expression is proper or improper. If improper, to rewrite it as the sum of a polynomial and a proper rational expression.

Answer to Problem 12AYU

Solution:

The rational expression is improper and proper rational expression is $2x + \frac{6x}{\left(x^2 + 1\right)}$.

Explanation of Solution

Given:

$\frac{2x\left(x^2 + 4\right)}{\left(x^2 + 1\right)}$.

Calculation:

By definition, a rational expression $\text{P/Q}$ is called proper if the degree of polynomial in the numerator is less than the degree of polynomial in the denominator; otherwise it is improper.

Here the degree of the polynomial in the numerator is 3 and the degree of the polynomial in the denominator is 2. Therefore, the given rational expression is improper.

It can be rewritten as the sum of polynomial and a proper rational expression as $2x + \frac{6x}{\left(x^2 + 1\right)}$.