# How do you divide ( x^5 - x^3 + 5x^2 - 10x - 75)/(x - 2 )?

Aug 6, 2017

The remainder is $\textcolor{red}{- 51}$ and the quotient is $= {x}^{4} + 2 {x}^{3} + 3 {x}^{2} + 11 x + 12$

#### Explanation:

Let's perform a synthetic division

$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a a a}$$|$$\textcolor{w h i t e}{a a}$$1$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a}$$5$$\textcolor{w h i t e}{a a a}$$- 10$$\textcolor{w h i t e}{a a a}$$- 75$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$____________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$2$$\textcolor{w h i t e}{a a a a a}$$4$$\textcolor{w h i t e}{a a a a a}$$6$$\textcolor{w h i t e}{a a a a}$$22$$\textcolor{w h i t e}{a a a a a}$$24$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$______________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$2$$\textcolor{w h i t e}{a a a a a}$$3$$\textcolor{w h i t e}{a a a a a}$$11$$\textcolor{w h i t e}{a a a}$$12$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{- 41}$

The remainder is $\textcolor{red}{- 51}$ and the quotient is $= {x}^{4} + 2 {x}^{3} + 3 {x}^{2} + 11 x + 12$

Therefore,

$\frac{{x}^{5} - {x}^{3} + 5 {x}^{2} - 10 x - 75}{x - 2}$

$= {x}^{4} + 2 {x}^{3} + 3 {x}^{2} + 11 x + 12 - \frac{11}{x - 2}$