Question

In the following exercises, perform indicated operations.

$\frac{-9p-17}{p^2-4p-21}-\frac{p+1}{7-p}$

Short answer

The resultant expression is $\frac{p+2}{p+3}$.

Step-by-step solution

Step 1. Given Information

The given expression is$\frac{-9p-17}{p^2-4p-21}-\frac{p+1}{7-p}$.

Step 2. Express with LCD

• The LCD of the given expressions to subtract is $\left(p-7\right)\left(p+3\right)$.
• Express the given expression with LCD.

$\begin{array}{rcl}\frac{-9p-17}{p^2-4p-21}-\frac{p+1}{7-p}& =& \frac{-9p-17}{\left(p-7\right)\left(p+3\right)}-\frac{-\left(p+1\right)\left(p+3\right)}{\left(p-7\right)\left(p+3\right)}\\ & =& \frac{-9p-17}{\left(p-7\right)\left(p+3\right)}-\frac{-p^2-4p-3}{\left(p-7\right)\left(p+3\right)}\end{array}$

Step 3. Simplify

• As the denominator is the same for both the terms, perform addition in the numerator and place it over the common denominator.

$\frac{-9p-17+p^2+4p+3}{\left(p-7\right)\left(p+3\right)}=\frac{p^2-5p+14}{\left(p-7\right)\left(p+3\right)}$

• Cancel out the common factor from numerator and denominator.

$\frac{\left(p-7\right)\left(p+2\right)}{\left(p-7\right)\left(p+3\right)}=\frac{p+2}{p+3}$

• So, the resultant expression is$\frac{p+2}{p+3}$.