Question

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

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

Short answer

The given expression is improper

Step-by-step solution

Step 1. Find the expression is proper or improper

The rational expression $\frac{P}{Q}$is proper if and only if the degree of the numerator is less than the degree of the denominator, otherwise it will be improper.

In the given rational expression $\frac{x^2+5}{x^2-4}$ , the degree of the numerator is equal to the degree of the denominator.

Step 2. Make the expression proper

To make the expression proper we have to perform long division in order to express the given expression as the sum of a polynomial and a proper rational expression.

That is,

$\begin{array}{r}\frac{1}{x^2-4\overline{)x^2+5}}\\ \frac{x^2-4}{9}\end{array}$

Therefore, $\frac{x^2+5}{x^2-4}=1+\frac{9}{x^2-4}$.

Now the expression is proper.