# How do you use synthetic division to divide X^3 + 2x^2 - 5x - 6 div (x + 3)?

Jul 5, 2018

The remainder is $0$ and the quotient is $= {x}^{2} - x - 2$

#### 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$$\textcolor{w h i t e}{a a a a a}$$2$$\textcolor{w h i t e}{a a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a a}$$- 6$

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{a a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a a a}$$6$

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

$\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}$$- 1$$\textcolor{w h i t e}{a a a a a a}$$- 2$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{red}{0}$

The remainder is $0$ and the quotient is $= {x}^{2} - x - 2$

Therefore,

$\frac{{x}^{3} + 2 {x}^{2} - 5 x - 6}{x + 3} = \left({x}^{2} - x - 2\right)$

Also,

If $f \left(x\right) = {x}^{3} + 2 {x}^{2} - 5 x - 6$

$f \left(- 3\right) = {\left(- 3\right)}^{3} + 2 {\left(- 3\right)}^{2} - 5 \left(- 3\right) - 6$

$= - 27 + 18 + 15 - 6$

$= 0$