## College Algebra (10th Edition)

$\displaystyle \frac{5x^{3}+2x-1}{x^{2}-4}$ is improper$,$ $\displaystyle \frac{5x^{3}+2x-1}{x^{2}-4}=5x+\frac{22x-1}{x^{2}-4}$
The rational expression $\displaystyle \frac{P}{Q}$ is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. Otherwise, the rational expression is called improper. --- The degree of the numerator, $P(x)=5x^{3}+2x-1, \quad$is $3$, and the degree of the denominator$, Q(x)=x^{2}-4, \quad$is $2$. The rational expression is improper. To make it proper, $\displaystyle \frac{5x^{3}+2x-1}{x^{2}-4} = \frac{5x(x^{2}-4)+20x+2x-1}{x^{2}-4}=\\\\\\ =\dfrac{5x(x^{2}-4)}{x^{2}-4}+\dfrac{22x-1}{x^{2}-4}$ $=5x+\displaystyle \frac{22x-1}{x^{2}-4}$