Question

In the following exercises, divide the rational expressions.

$\frac{{\displaystyle \frac{4d^2+7d-2}{35d+10}}}{{\displaystyle \frac{d^2-4}{7d^2-12d-4}}}$

Short answer

The simplified expression is$\frac{4d-1}{5}.$

Step-by-step solution

Step 1. Given Information

$\frac{{\displaystyle \frac{4d^2+7d-2}{35d+10}}}{{\displaystyle \frac{d^2-4}{7d^2-12d-4}}}=\frac{4d^2+7d-2}{35d+10}÷\frac{d^2-4}{7d^2-12d-4}$

Step 2. Rewrite as product of first times reciprocalof second.

$\frac{4d^2+7d-2}{35d+10}÷\frac{d^2-4}{7d^2-12d-4}=\frac{4d^2+7d-2}{35d+10}\times \frac{7d^2-12d-4}{d^2-4}$

Step 3. Factor the numerators and thedenominators, and then multiply.

$$\begin{gathered} =\frac{\left(4d-1\right)\left(d+2\right)}{5\left(7d+2\right)}\times \frac{\left(d-2\right)\left(7d+2\right)}{\left(d-2\right)\left(d+2\right)} \\ =\frac{\left(4d-1\right)\left(d+2\right)\left(d-2\right)\left(7d+2\right)}{5\left(7d+2\right)\left(d-2\right)\left(d+2\right)} \end{gathered}$$

Step 4. Simplify by dividing out common factors. 

$=\frac{4d-1}{5}$