## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$\text{improper};$ $\dfrac{x^{2}+5}{x^{2}-4}=1+\dfrac{9}{x^{2}-4}$
A rational expression $\dfrac{A(x)}{B(x)}$ is said to be proper if the degree of polynomial in numerator is less than the degree of its denominator.If it does not happen , then it is said to be improper rational polynomial. Here, the degree of the numerator $A(x)= x^2+5$ is $2$ and the degree of the denominator ; $B(x)=x^2-4$ is $2$. We see that the given rational expression is improper. To make the rational expression proper, we will solve as: $\dfrac{x^{2}+5}{x^{2}-4}$=$\displaystyle \frac{x^{2}-4+9}{x^{2}-4} \\ =\dfrac{x^{2}-4}{x^{2}-4}+\dfrac{9}{x^{2}-4} \\=1+\dfrac{9}{x^{2}-4}$