# How do you divide using synthetic division: (2u^4 - 5u^3 - 12u^2 + 2u - 8)/(u - 4)?

##### 2 Answers
Oct 19, 2017

$\frac{2 {u}^{4} - 5 {u}^{3} - 12 {u}^{2} + 2 u - 8}{u - 4} = \textcolor{red}{2 {u}^{3} + 3 {u}^{2} + 2}$
with a Remainder of $\textcolor{b l u e}{0}$
$\textcolor{w h i t e}{\text{XXX}}$(see below for solution method using synthetic division)

#### Explanation:

{: (,,,color(grey)(u^4),color(grey)(u^3),color(grey)(u^2),color(grey)(u^1),color(grey)(u^0)), (,," | ",2,-5,-12,+2,-8), (,ul(+color(white)("xxx"))," | ",ul(color(white)(0)),ul(+8),ul(+12),ul(+0),ul(+8)), (,xxcolor(magenta)4," | ",color(red)2,color(red)(+3),color(white)(+0)color(red)(0),color(white)("+")color(red)(2),color(white)("+")0), (,,,color(grey)(u^3),color(grey)(u^2),color(white)(+0)color(grey)(u^1),color(white)("+")color(grey)(u^0)color(white)("+"),color(blue)("R")) :}

Rows  and  are not really part of the synthetic division; they are here for reference purposes only.

Row  are the coefficients of the variables in row 

Values in Row  are the sum of the values in the same column from Rows  and 

Values in Row  are the product of the values from the previous column of Row  and $\textcolor{m a \ge n t a}{4}$ where $\textcolor{m a \ge n t a}{4}$ is the value of $u$ necessary to make the divisor $\left(u - 4\right)$ equal to $0$

Oct 19, 2017

The remainder is $\textcolor{red}{0}$ and the quotient is $= 2 {u}^{3} + 3 {u}^{2} + 2$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a}$$4$$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$2$$\textcolor{w h i t e}{a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a a a}$$- 12$$\textcolor{w h i t e}{a a a a a}$$2$$\textcolor{w h i t e}{a a a a a a}$$- 8$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$- - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$8$$\textcolor{w h i t e}{a a a a a a a}$$12$$\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}$$8$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$- - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a}$$2$$\textcolor{w h i t e}{a a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$2$$\textcolor{w h i t e}{a a a a a a a a}$$\textcolor{red}{0}$

The remainder is $\textcolor{red}{0}$ and the quotient is $= 2 {u}^{3} + 3 {u}^{2} + 0 u + 2$

$\frac{2 {u}^{4} - 5 {u}^{3} - 12 {u}^{2} + 2 u - 8}{u - 4} = 2 {u}^{3} + 3 {u}^{2} + 0 u + 2$