Chapter 11.5, Problem 11AYU

In Problems 5-12, 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. $\frac{x\left(x-1\right)}{\left(x+4\right)\left(x-3\right)}$

To determine

To find: The given rational expression is proper or improper. If improper, to rewrite it as the sum of a polynomial and a proper rational expression.

Answer to Problem 11AYU

Solution:

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

Explanation of Solution

Given:

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

Calculation:

By definition, a rational expression $\frac{P}{Q}$ is called proper if the degree of polynomial in the numerator is less than the degree of polynomial in the denominator; otherwise it is improper.

Here the degree of the polynomial in the numerator is 2 and the degree of the polynomial in the denominator is also 2. Therefore, the given rational expression is improper.

It can be rewritten as the sum of polynomial and a proper rational expression as $1 + \frac{ - 2\left(x - 6\right)}{\left(x + 4\right)\left(x - 3\right)}$.

$1+\frac{-2\left(x-6\right)}{\left(x+4\right)\left(x-3\right)}=\frac{\left(x+4\right)\left(x-3\right)-2\left(x-6\right)}{\left(x+4\right)\left(x-3\right)}=\frac{x\left(x-1\right)}{\left(x+4\right)\left(x-3\right)}$