# How do you divide x^2/(2x + 4)?

Apr 21, 2017

There are several methods. I will try to show you long division.

#### Explanation:

Write the dividend with 0s for the missing terms:

$\textcolor{w h i t e}{\frac{2 x + 4}{\textcolor{b l a c k}{2 x + 4}}} \frac{\textcolor{w h i t e}{\left({x}^{2} + 0 x + 0\right)}}{\text{)} \textcolor{w h i t e}{x} {x}^{2} + 0 x + 0}$

Please observe that ${x}^{2} / \left(2 x\right) = \frac{1}{2} x$, therefore, we put $\frac{1}{2} x$ in the quotient:

$\textcolor{w h i t e}{\frac{2 x + 4}{\textcolor{b l a c k}{2 x + 4}}} \frac{\frac{1}{2} x \textcolor{w h i t e}{0 x + 0}}{\text{)} \textcolor{w h i t e}{x} {x}^{2} + 0 x + 0}$

We multiply $\frac{1}{2} x \left(2 x + 4\right) = {x}^{2} + 2 x$ and then we subtract this underneath:

$\textcolor{w h i t e}{\frac{2 x + 4}{\textcolor{b l a c k}{2 x + 4}}} \frac{\frac{1}{2} x \textcolor{w h i t e}{0 x + 0}}{\text{)} \textcolor{w h i t e}{x} {x}^{2} + 0 x + 0}$
$\textcolor{w h i t e}{\text{.............}} \underline{- {x}^{2} - 2 x}$
$\textcolor{w h i t e}{\text{.....................}} - 2 x$

Please observe that $\frac{- 2 x}{2 x} = - 1$, therefore, we add $- 1$ in the quotient:

$\textcolor{w h i t e}{\frac{2 x + 4}{\textcolor{b l a c k}{2 x + 4}}} \frac{\frac{1}{2} x - 1 \textcolor{w h i t e}{+ 0}}{\text{)} \textcolor{w h i t e}{x} {x}^{2} + 0 x + 0}$
$\textcolor{w h i t e}{\text{.............}} \underline{- {x}^{2} - 2 x}$
$\textcolor{w h i t e}{\text{.....................}} - 2 x$

We multiply $- 1 \left(2 x + 4\right) = - 2 x - 4$ and then we subtract this underneath:

$\textcolor{w h i t e}{\frac{2 x + 4}{\textcolor{b l a c k}{2 x + 4}}} \frac{\frac{1}{2} x - 1 \textcolor{w h i t e}{+ 0}}{\text{)} \textcolor{w h i t e}{x} {x}^{2} + 0 x + 0}$
$\textcolor{w h i t e}{\text{.............}} \underline{- {x}^{2} - 2 x}$
$\textcolor{w h i t e}{\text{.....................}} - 2 x + 0$
$\textcolor{w h i t e}{\text{........................}} \underline{2 x + 4}$
color(white)("................................")4" " larr This is the remainder.

You can write the remainder as $\frac{4}{2 x + 4}$ or $\frac{2}{x + 2}$

The results of the division is:

x^2/(2x + 4)= 1/2x-1+2/(x+2