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.

$f\left(x\right)=\frac{4x}{x^2-3x-10}andg\left(x\right)=\frac{x^2-25}{8x^2}$

Short answer

The value of$R\left(x\right)=\frac{\left(x+5\right)}{2x\left(x+2\right)}$

Step-by-step solution

Step 1. Given Information

$f\left(x\right)=\frac{4x}{x^2-3x-10}andg\left(x\right)=\frac{x^2-25}{8x^2}$

Step 2. Factor each numerator and denominator

$$\begin{gathered} R\left(x\right)=\frac{4x}{x^2-3x-10}.\frac{x^2-25}{8x^2} \\ R\left(x\right)=\frac{4x}{\left(x-5\right)\left(x+2\right)}.\frac{\left(x-5\right)\left(x+5\right)}{8x\times x} \\ R\left(x\right)=\frac{4x\left(x-5\right)\left(x+5\right)}{8x.x\left(x-5\right)\left(x+2\right)} \end{gathered}$$

Step 3. Remove common factors and simplify

$R\left(x\right)=\frac{\left(x+5\right)}{2x\left(x+2\right)}$