Question

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

$\frac{4q^2-q+3}{q^2+6q+5}-\frac{3q^2+q+6}{q^2+6q+5}.$

Short answer

The answer is$\frac{q-3}{q+5}.$

Step-by-step solution

Step 1. Given Information.

Given:

$\frac{4q^2-q+3}{q^2+6q+5}-\frac{3q^2+q+6}{q^2+6q+5}$having same denominator.

Step 2. Solving the given expression.

$$\begin{gathered} Asthedenominatorissamesowecandirectlysubtractthenumerators: \\ \frac{4q^2-q+3}{q^2+6q+5}-\frac{3q^2+q+6}{q^2+6q+5} \\ =\frac{4q^2-q+3-(3q^2+q+6)}{q^2+6q+5} \\ =\frac{4q^2-q+3-3q^2-q-6}{q^2+6q+5} \\ =\frac{q^2-2q-3}{q^2+6q+5} \\ =\frac{(q+1)(q-3)}{(q+1)(q+5)}=\frac{q-3}{q+5}. \end{gathered}$$