Chapter 5.7, Problem 4MRPS

Solution

To Find: Two rational expressions $r\left(x\right)\text{ and }s\left(x\right)$ each containing a quadratic polynomial.

The required rational expressions are $r\left(x\right)=\frac{x^2-5x+6}{x^2+5x+6}$ and $s\left(x\right)=\frac{x^2+5x+6}{x^2+2x-8}$.

Given information:

  $r\left(x\right)\text{ and }s\left(x\right)$ each contain a quadratic polynomial and $r\left(x\right)\cdot s\left(x\right)=\frac{x-3}{x+4}$.

Explanation:

Let us assume the following:

  $x-3$ is one of the factors of the numerator of $r\left(x\right)$ and $x+4$ is one of the factors of the denominator of $s\left(x\right)$.

The other factor of the numerator of $r\left(x\right)$ is same as the other factor of the denominator of $s\left(x\right)$ .

The factors of the denominator of $r\left(x\right)$ is same as the factors of the numerator of $s\left(x\right)$ .

So, if we take $r\left(x\right)=\frac{x^2-5x+6}{x^2+5x+6}$ and $s\left(x\right)=\frac{x^2+5x+6}{x^2+2x-8}$ , then

  $\begin{array}{c}r\left(x\right)\cdot s\left(x\right)=\frac{x^2-5x+6}{x^2+5x+6}\times \frac{x^2+5x+6}{x^2+2x-8}\\ =\frac{\left(x-3\right)\left(x-2\right)}{\left(x+3\right)\left(x+2\right)}\times \frac{\left(x+2\right)\left(x+3\right)}{\left(x-2\right)\left(x+4\right)}\left(\text{Factorise the numerators and denomitaors}\right)\\ =\frac{x-3}{x+4}\left(\text{Divide out the common factors}\right)\end{array}$

Therefore, $r\left(x\right)=\frac{x^2-5x+6}{x^2+5x+6}$ and $s\left(x\right)=\frac{x^2+5x+6}{x^2+2x-8}$.