Question

In the following exercises, add or subtract.

$\frac{c^2+5c-10}{c^2-16}-\frac{c^2-8c-10}{16-c^2}$

Short answer

The difference of given rational expressions is$\frac{2c+10}{c+4}$.

Step-by-step solution

Step 1. Given Information

The given expression is $\frac{c^2+5c-10}{c^2-16}-\frac{c^2-8c-10}{16-c^2}$

Step 2. Equalise the denominators

• Multiply the numerator and denominator of the second term by $-1$ to make both the numerators equal.

$\frac{c^2+5c-10}{c^2-16}-\frac{c^2-8c-10}{16-c^2}=\frac{c^2+5c-10}{c^2-16}-\frac{-c^2+8c+10}{c^2-16}$

Step 3. Subtract the numerators  

• As the denominators of the rational expressions is equal, subtract the numerators and place them over a common denominator.

$\begin{array}{rcl}\frac{c^2+5c-10}{c^2-16}-\frac{-c^2+8c+10}{c^2-16}& =& \frac{c^2+5c-10-\left(-c^2+8c+10\right)}{c^2-16}\\ & =& \frac{c^2+5c-10+c^2-8c-10}{c^2-16}\\ & =& \frac{2c^2-3c-20}{c^2-16}\end{array}$

Step 4. Factor the numerator and denominator  

• Factor the expression in the numerator and denominator and then cancel out the common terms in numerator and denominator .

$\begin{array}{rcl}\frac{2c^2-3c-20}{c^2-16}& =& \frac{2\cancel{\left(c-4\right)}\left(c+5\right)}{\cancel{\left(c-4\right)}\left(c+4\right)}\\ & =& \frac{2c+10}{c+4}\end{array}$