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

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:

{: ([0],,,color(grey)(u^4),color(grey)(u^3),color(grey)(u^2),color(grey)(u^1),color(grey)(u^0)), ([1],," | ",2,-5,-12,+2,-8), ([2],ul(+color(white)("xxx"))," | ",ul(color(white)(0)),ul(+8),ul(+12),ul(+0),ul(+8)), ([3],xxcolor(magenta)4," | ",color(red)2,color(red)(+3),color(white)(+0)color(red)(0),color(white)("+")color(red)(2),color(white)("+")0), ([4],,,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 [0] and [4] are not really part of the synthetic division; they are here for reference purposes only.

Row [1] are the coefficients of the variables in row [0]

Values in Row [3] are the sum of the values in the same column from Rows [1] and [2]

Values in Row [2] are the product of the values from the previous column of Row [3] 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 $= 2 {u}^{3} + 3 {u}^{2} + 0 u + 2$

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