Q. 8 - Precalculus Enhanced with Graphing Utilities

Question

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.

$\frac{3x^2-2}{x^2-1}$.

Step-by-Step Solution

Answer

The rational expression $\frac{3x^2-2}{x^2-1}$ is improper.

The sum of a polynomial and a proper rational expression is $3+\frac{1}{x^2-1}$.

Step 1. Given Information.

The rational expression is,

$\frac{3x^2-2}{x^2-1}$.

Step 2. Proper or Improper rational expression.

$\frac{3x^2-2}{x^2-1}$

The degree of the numerator$=2$.

The degree of the denominator$=2$

The degree of the numerator$=$The degree of the denominator.

It is an improper rational expression.

Step 3. Sum of polynomial and Proper rational expression.

The improper rational expression is,

$\frac{3x^2-2}{x^2-1}$

Add and subtract $1$ in the numerator,

$$\begin{gathered} =\frac{3x^2-2+1-1}{x^2-1} \\ =\frac{3x^2-3+1}{x^2-1} \\ =\frac{3x^2-3}{x^2-1}+\frac{1}{x^2-1} \\ =\frac{3(x^2-1)}{x^2-1}+\frac{1}{x^2-1} \\ =3+\frac{1}{x^2-1} \end{gathered}$$

The polynomial is $3$ and the proper rational expression is $\frac{1}{x^2+1}$.