Question

Given $f\left(x\right)=\frac{x-4}{x^2-3x-10}$and $g\left(x\right)=\frac{x-5}{x^2-2x-8}$.

Find$R\left(x\right)=f\left(x\right)÷g\left(x\right)$.

Short answer

The solution is$\frac{\left(x-4\right)\left(x-4\right)}{\left(x-5\right)\left(x-5\right)}$.

Step-by-step solution

Step 1. Given information.

The given functions are$f\left(x\right)=\frac{x-4}{x^2-3x-10}$and$g\left(x\right)=\frac{x-5}{x^2-2x-8}$.

Step 2. Simplify the expression.

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

Step 3. Simplified answer.

Hence, the simplified expression is$R\left(x\right)=\frac{\left(x-4\right)\left(x-4\right)}{\left(x-5\right)\left(x-5\right)}$.