How do you simplify (5(1-b)+15)/(b^2-16)?

Oct 18, 2015

$- \frac{5}{b + 4}$

Explanation:

There are some inteesting techniques to use to simplify this expression.

First, start by focusing on the denominator. Notice that $16$ is a perfect square

#16 = 4 * 4 = 4^2

which means that you're dealing with the difference of two squares

$\textcolor{b l u e}{{a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)}$

In this case, you would have

${b}^{2} - 16 = {b}^{2} - {4}^{2} = \left(b - 4\right) \left(b + 4\right)$

Now focus on the numerator. Notice that you can use $5$ as a common factor for the two terms

$5 \left(1 - b\right) + 15 = 5 \cdot \left[\left(1 - b\right) + 3\right] = 5 \cdot \left(4 - b\right)$

Now, you can change the sign of the terms by recognizing that

$4 - b = - \left(b - 4\right)$

The numerator will thus be equivalent to

$5 \left(1 - b\right) + 15 = - 5 \cdot \left(b - 4\right)$

The expression will be

$\frac{- 5 \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(b - 4\right)}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(b - 4\right)}}} \left(b + 4\right)} = \textcolor{g r e e n}{- \frac{5}{b + 4}}$