# How do you use synthetic division to divide x^3 + 2x^2 - 2x + 24 by x+4?

May 18, 2018

$\frac{{x}^{3} + 2 {x}^{2} - 2 x + 24}{x + 4} = {x}^{2} - 2 x + 6$

#### Explanation:

Divide ${x}^{3}$ by $x + 4$
Quotient $= {x}^{2}$
Remainder $= {x}^{3} + 2 {x}^{2} - 2 x + 24 - {x}^{3} - 4 {x}^{2}$
Remainder $= - 2 {x}^{2} - 2 x + 24$

Divide $- 2 {x}^{2}$ by $x + 4$
Quotient $= - 2 x$
Remainder $= - 2 {x}^{2} - 2 x + 24 + 2 {x}^{2} + 8 x$
Remainder $= 6 x + 24$

Divide $6 x$ by $x + 4$
Quotient $= 6$
Remainder $= 0$

May 18, 2018

Synthetic division gives: ${x}^{2} - 2 x + 6$

Process shown in detail

#### Explanation:

Given: $\frac{{x}^{3} + 2 {x}^{2} - 2 x + 24}{x + 4}$

Consider the denominator $x + 4$

Set $x + 4 = 0 \implies \textcolor{red}{x = - 4}$

Now we construct the coefficient manipulation.

$\textcolor{w h i t e}{\text{dddd}} {x}^{3} + 2 {x}^{2} - 2 x + 24$
$\textcolor{w h i t e}{\text{ddd")darrcolor(white)("d.d") darrcolor(white)("ddd")darrcolor(white)("ddd}} \downarrow$

color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
bring down the first value of $\textcolor{m a \ge n t a}{1}$

color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24
color(white)("d..d")ul(|color(magenta)(darr )
$\textcolor{w h i t e}{\text{dddd}} \textcolor{m a \ge n t a}{1}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$- 4 \times \textcolor{m a \ge n t a}{1} = \textcolor{b l u e}{- 4}$
Place the $\textcolor{b l u e}{- 4}$ under the 2

color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24
color(white)("d..d")ul(|color(magenta)(darr )color(white)("d")color(blue)(-4) )
$\textcolor{w h i t e}{\text{dddd}} \textcolor{m a \ge n t a}{1}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add the 2 and the $\textcolor{b l u e}{- 4}$

color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24
color(white)("d..d")ul(|color(magenta)(darr )color(white)("d")color(blue)(-4) )
$\textcolor{w h i t e}{\text{dddd")color(magenta)(1)color(white)("d}} - 2$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{red}{- 4} \times \text{ the new } - 2 = + 8$
Place the $- 8$ under the $- 2$

color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24
color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8 )
$\textcolor{w h i t e}{\text{dddd")1color(white)("dd}} - 2$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add the $- 2 \text{ and the } 8$

color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24
color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8 )
$\textcolor{w h i t e}{\text{dddd")1color(white)("dd")-2color(white)("ddddd}} 6$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{red}{- 4} \times \text{the new } 6 = - 24$
Place the -24 under the existing +24
Add and you get the 0 so it is an exact division. No remainder.

color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24
color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8color(white)("ddd")24 )
$\textcolor{w h i t e}{\text{dddd")1color(white)("dd")-2color(white)("ddddd")6color(white)("ddd}} 0$
$\textcolor{w h i t e}{\text{ddd.")darrcolor(white)("ddd")darrcolor(white)("dddd}} \downarrow$

$\textcolor{w h i t e}{\text{dddd")x^2color(white)("d")-2xcolor(white)("dd}} + 6$