Question

In the following exercises, multiply the rational expressions.

$\frac{2n^2-3n-14}{25-n^2}.\frac{n^2-10n+25}{2n^2-13n+21}$

Short answer

The simplified rational expression is$\frac{-\left(n+2\right)\left(n-5\right)}{\left(n+5\right)\left(n-3\right)}$

Step-by-step solution

Step 1. Given Information

$\frac{2n^2-3n-14}{25-n^2}.\frac{n^2-10n+25}{2n^2-13n+21}$

We have to multiply the given rational expression.

Step 2. Factor the numerators and denominators

$$\begin{gathered} =\frac{\left(2n-7\right)\left(n+2\right)}{\left(5+n\right)\left(5-n\right)}.\frac{\left(n-5\right)\left(n-5\right)}{\left(2n-7\right)\left(n-3\right)} \\ =\frac{\left(2n-7\right)\left(n+2\right)\left(n-5\right)\left(n-5\right)}{-\left(5+n\right)\left(n-5\right)\left(2n-7\right)\left(n-3\right)} \end{gathered}$$

Step 3. Remove the common factors and simplify

$=\frac{-\left(n+2\right)\left(n-5\right)}{\left(n+5\right)\left(n-3\right)}$