Q. 11 - 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.

$\frac{x(x-1)}{(x+4)(x-3)}$.

Step-by-Step Solution

Answer

The rational expression $\frac{x(x-1)}{(x+4)(x-3)}$is improper.

The sum of a polynomial and a proper rational expression is $1+\frac{-2(x-6)}{(x+4)(x-3)}$.

Step 1. Given Information.

The rational expression is,

$\frac{x(x-1)}{(x+4)(x-3)}$.

Step 2. Simplifying the rational expression.

The rational expression is simplified,

$$\begin{gathered} \frac{x(x-1)}{(x+4)(x-3)} \\ =\frac{x^2-x}{x^2-3x+4x-12} \\ =\frac{x^2-x}{x^2+x-12} \end{gathered}$$

Step 3. Improper or Proper rational expressions.

$\frac{x^2-x}{x^2+x-12}$

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 4. Decomposition of the rational expression.

The improper rational expression is,

$\frac{x^2-x}{x^2+x-12}$,

Dividing the expression, we get,

$\frac{x^2-x}{x^2+x-12}=1+\frac{-2x+12}{x^2+x-12}$

                    $=1+\frac{-2(x-6)}{x^2+x-12}$