# How do you find f(-3) given f(x)=x^3-2x^2+4x+6 using synthetic division?

Aug 5, 2017

The remainder is $\textcolor{red}{- 51}$ and the quotient is $= {x}^{2} - 5 x + 19$

#### 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 a}$$|$$\textcolor{w h i t e}{a a}$$1$$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{a a a a a a}$$4$$\textcolor{w h i t e}{a a a a a a a}$$6$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$_________

$\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}$$- 3$$\textcolor{w h i t e}{a a A a a}$$15$$\textcolor{w h i t e}{a a a a a}$$- 57$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a a a}$$19$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{- 51}$

The remainder is $\textcolor{red}{- 51}$ and the quotient is $= {x}^{2} - 5 x + 19$

Therefore,

$\frac{{x}^{3} - 2 {x}^{2} + 4 x + 6}{x + 3} = {x}^{2} - 5 x + 19 - \frac{51}{x + 3}$