# How do you use the remainder theorem to determine the remainder when the polynomial 2x^4-2x^3+5x is divided by x+2?

Feb 28, 2017

The remainder is $= 38$

#### Explanation:

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

$\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}$$- 4$$\textcolor{w h i t e}{a a a a}$$12$$\textcolor{w h i t e}{a a a}$$- 24$$\textcolor{w h i t e}{a a a a a a}$$38$

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

$\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}$$2$$\textcolor{w h i t e}{a a a a a}$$- 6$$\textcolor{w h i t e}{a a a a}$$12$$\textcolor{w h i t e}{a a a}$$- 19$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{red}{38}$

The remainder is $= 38$

The quotient is $= 2 {x}^{3} - 6 {x}^{2} + 12 x - 19$

Verification

Using the remainder theorem

$f \left(- 2\right) = 32 + 16 - 10 = 38$