# Factorization of Fractions

You can simplify fractions through factorization to make calculations—and your life—easier.

Example 1

Simplify the fraction $\frac{2x+4}{x+2}$

Here you should factorize the numerator and denominator separately. If you find the same factors above and below the fraction bar, they can be canceled. The numerator is

 $2x+4=2×x+2×2=2\left(x+2\right)$

The denominator is just $x+2$. You can now see that $x+2$ is a common factor for the numerator and denominator, and you therefore have that

 $\frac{2x+4}{x+2}=\frac{2\text{(x+2)}}{\text{x+2}}=2$

Here is an example with powers:

Example 2

Simplify the fraction $\frac{2{b}^{3}+4{b}^{6}}{12+24{b}^{3}}$

You factorize the numerator first and get

 $2{b}^{3}+4{b}^{6}=2{b}^{3}\phantom{\rule{-0.17em}{0ex}}\left(1+2{b}^{3}\right)$

Then you factorize the denominator and get

 $12+24{b}^{3}=12\phantom{\rule{-0.17em}{0ex}}\left(1+2{b}^{3}\right)$

You can see that $1+2{b}^{3}$ is a common factor in the numerator and denominator, and can therefore be canceled. You get

$\begin{array}{llll}\hfill \frac{2{b}^{3}+4{b}^{6}}{12+24{b}^{3}}& =\frac{2{b}^{3}\text{(1+2b3)}}{12\text{(1+2b3)}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2{b}^{3}}{12}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{{b}^{3}}{6}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $\frac{2{b}^{3}+4{b}^{6}}{12+24{b}^{3}}=\frac{2{b}^{3}\text{(1+2b3)}}{12\text{(1+2b3)}}=\frac{2{b}^{3}}{12}=\frac{{b}^{3}}{6}$