# How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems x^5 + 2x^4 - 3x + 3 divided by x - 1?

Feb 19, 2017

The remainder is $= 3$

#### Explanation:

The remainder theorem states that when a polynomial $f \left(x\right)$ is divided by $x - c$

$f \left(x\right) = \left(x - c\right) q \left(x\right) + r \left(x\right)$

$f \left(c\right) = 0 + r$

Here,

$f \left(x\right) = {x}^{5} + 2 {x}^{4} - 3 x + 3$

and $\left(x - 1\right)$

$f \left(1\right) = 1 + 2 - 3 + 3 = 3$

The remainder is $= 3$

We now 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}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{a a a a}$$3$

$\textcolor{w h i t e}{a a a a}$color(white)(aaaaa)|$\textcolor{w h i t e}{a a a a a}$color(white)(aaaa)1$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a a}$$0$

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

$\textcolor{w h i t e}{a a a a}$color(white)(aaaaa)$\textcolor{w h i t e}{a a a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{3}$

The remainder is also $= 3$

The quotient is $= {x}^{4} + 3 {x}^{3} + 3 {x}^{2} + 3 x$