# How do you use synthetic division to find x=-1 for P(x)= x^3+2x^2-5x+6 ?

Jun 6, 2017

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

#### Explanation:

Let's perform the synthetic division

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

$\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}$$- 1$$\textcolor{w h i t e}{a a a a a a}$$- 1$$\textcolor{w h i t e}{a a a a a}$$6$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$________

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

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

$\frac{{x}^{3} + 2 {x}^{2} - 5 x + 6}{x + 1} = {x}^{2} + x - 6 + \frac{12}{x + 1}$