# How do you use synthetic division to divide ( x^4 + 3x^3 - 5x^2 + 9 ) / ( x + 3 )?

Jul 4, 2018

The remainder is $= - 37$ and the quotient is ${x}^{3} - 5 x + 15$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$- 3$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$9$

$\textcolor{w h i t e}{a a a a a a a}$|color(white)(aaaa)color(white)(aaa)-3$\textcolor{w h i t e}{a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$15$$\textcolor{w h i t e}{a a a a}$$- 45$

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

$\textcolor{w h i t e}{a a a a a a a}$|color(white)(aaaa)$1$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a}$$15$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{- 36}$

The remainder is $= - 37$ and the quotient is ${x}^{3} - 5 x + 15$

$\frac{{x}^{4} + 3 {x}^{3} - 5 {x}^{2} + 0 x + 9}{x + 3} = \left({x}^{3} - 5 x + 15\right) - \frac{36}{x + 3}$