## Precalculus (10th Edition)

Improper; Proper Form: $3+\frac{1}{x^2-1}$
A rational expression is proper if and only if the degree of its numerator's polynomial is less than the degree of its denominator's polynomial. Here, the degree of the numerator is $2$ while the degree of the denominator is also $2$ Hence, the rational expression is improper. Make the rational expression proper by rewriting then simplify: \begin{align*} \frac{3x^2-2}{x^2-1}&=\frac{3x^2-3+1}{x^2-1}\\ &=\dfrac{3(x^2-1)+1}{x^2-1}\\ &=\frac{3(x^2-1)}{x^2-1}+\frac{1}{x^2-1}\\ &=3+\frac{1}{x^2-1} \end{align*}