Question

For the following exercises, find $R\left(x\right)=f\left(x\right)g\left(x\right)$where $f\left(x\right)andg\left(x\right)$are given.

$f\left(x\right)=\frac{x^2-2x}{x^2+6x-16}andg\left(x\right)=\frac{x^2-64}{x^2-8x}$

Short answer

The value of$R\left(x\right)=1$

Step-by-step solution

Step 1. Given Information

$f\left(x\right)=\frac{x^2-2x}{x^2+6x-16}andg\left(x\right)=\frac{x^2-64}{x^2-8x}$

Step 2. Factor each numerator and denominator

$$\begin{gathered} R\left(x\right)=\frac{x^2-2x}{x^2+6x-16}.\frac{x^2-64}{x^2-8x} \\ R\left(x\right)=\frac{x\left(x-2\right)}{\left(x+8\right)\left(x-2\right)}.\frac{\left(x-8\right)\left(x+8\right)}{x\left(x-8\right)} \end{gathered}$$

Step 3. Remove common factors and simplify

$$\begin{gathered} R\left(x\right)=\frac{x\left(x-2\right)\left(x-8\right)\left(x+8\right)}{x\left(x-8\right)\left(x+8\right)\left(x-2\right)} \\ R\left(x\right)=1 \end{gathered}$$