# How do you divide (x^4-x^3+ 3 x^2-4x-2)/(x+2) ?

Jul 15, 2018

The remainder is $42$ and the quotient is $= {x}^{3} - 3 {x}^{2} + 9 x - 22$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$- 2$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a a}$$- 4$$\textcolor{w h i t e}{a a a a}$$- 2$

$\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$$\textcolor{w h i t e}{a a a a a a}$$6$$\textcolor{w h i t e}{a a a a a}$$- 18$$\textcolor{w h i t e}{a a a a}$$44$

$\textcolor{w h i t e}{a a a a a a a a a}$_________________________________________________________

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{a a a a a a}$$9$$\textcolor{w h i t e}{a a a a a}$$- 22$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{42}$

The remainder is $42$ and the quotient is $= {x}^{3} - 3 {x}^{2} + 9 x - 22$

$\frac{{x}^{4} - {x}^{3} + 3 {x}^{2} - 4 x - 2}{x + 2} = {x}^{3} - 3 {x}^{2} + 9 x - 22 + \frac{42}{x + 2}$