# How do you evaluate f(x)=-x^4+8x^3+13x-4 at x=-2 using direct substitution and synthetic division?

Feb 24, 2017

The answer is $= - 110$

#### Explanation:

Direct substitution

$f \left(- 2\right) = - 16 - 64 - 26 - 4 = - 110$

The remainder is $= - 110$

We perform a synthetic division

$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{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}$$8$$\textcolor{w h i t e}{a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$13$$\textcolor{w h i t e}{a a a a}$$- 4$

$\textcolor{w h i t e}{a a a a a a}$$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$\textcolor{w h i t e}{a a a a a a a}$$2$$\textcolor{w h i t e}{a a a a a}$$- 20$$\textcolor{w h i t e}{a a a a}$$40$$\textcolor{w h i t e}{a a a a a}$$- 106$

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

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{w h i t e}{a a a a a a}$$- 1$$\textcolor{w h i t e}{a a a}$$10$$\textcolor{w h i t e}{a a a a a}$$- 20$$\textcolor{w h i t e}{a a a a a}$$53$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{- 110}$

The remainder is $= - 110$

The quotient is $= - {x}^{3} + 10 {x}^{2} - 20 x + 53$