# How do you divide (n^4+5n^3-6n+3)div(n+3) using synthetic division?

May 30, 2017

The remainder is $= - 33$ and the quotient is $= {n}^{3} + 2 {n}^{2} - 6 n + 12$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{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 a a}$$5$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$- 6$$\textcolor{w h i t e}{a a a a a a a}$$3$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$_________

$\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 a}$$- 3$$\textcolor{w h i t e}{a a a a a a}$$- 6$$\textcolor{w h i t e}{a a a a}$$18$$\textcolor{w h i t e}{a a a a a}$$- 36$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a a a a}$$2$$\textcolor{w h i t e}{a a a a a a}$$- 6$$\textcolor{w h i t e}{a a a a}$$12$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{- 33}$

$\frac{{n}^{4} + 5 {n}^{3} - 6 n + 3}{n + 3} = \left({n}^{3} + 2 {n}^{2} - 6 n + 12\right) - \frac{33}{n + 3}$

The remainder is $= - 33$ and the quotient is $= {n}^{3} + 2 {n}^{2} - 6 n + 12$