# How do you find f(3) given f(x)=x^3-7x^2+5x-6 using synthetic division?

Jul 3, 2018

The remainder is $- 27$ and the quotient is $= {x}^{2} - 4 x - 7$

#### 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$$\textcolor{w h i t e}{a a a a}$$- 7$$\textcolor{w h i t e}{a a a a a a}$$5$$\textcolor{w h i t e}{a a a a a a}$$- 6$

$\textcolor{w h i t e}{a a a a a}$|color(white)(aaaa)color(white)(aaaaaa)3$\textcolor{w h i t e}{a a a a a}$$- 12$$\textcolor{w h i t e}{a a a a}$$- 21$

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

$\textcolor{w h i t e}{a a a a a}$|color(white)(aaaa)$1$$\textcolor{w h i t e}{a a a a}$$- 4$$\textcolor{w h i t e}{a a a a a a}$$- 7$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{red}{- 27}$

The remainder is $- 27$ and the quotient is $= {x}^{2} - 4 x - 7$

Therefore,

$\frac{{x}^{3} - 7 {x}^{2} + 5 x - 6}{x - 3} = \left({x}^{2} - 4 x - 7\right) - \frac{27}{x - 3}$