# How do you divide (x^4-8x^2+16)div(x+2) using synthetic division?

Sep 3, 2017

$\left({x}^{4} - 8 {x}^{2} + 16\right) \div \left(x + 2\right) = \textcolor{red}{1 {x}^{3} - 2 {x}^{2} - 4 x + 8}$
with a Remainder of $\textcolor{g r e e n}{0}$

#### Explanation:

${x}^{4} - 8 {x}^{2} + 16 = \textcolor{b l u e}{1} {x}^{4} \textcolor{b l u e}{+ 0} {x}^{3} \textcolor{b l u e}{- 8} {x}^{2} \textcolor{b l u e}{+ 0} {x}^{1} \textcolor{b l u e}{+ 16} {x}^{0}$

To divide by $\left(x + 2\right)$ we perform synthetic substitution with x=color(magenta)(""(-2))

{: (,color(white)("xxx"),color(grey)(x^4),color(grey)(x^3),color(grey)(x^2),color(grey)(x^1),color(grey)(x^0),color(white)("xxx"),"row 0"), (,,color(blue)1,color(blue)(+0),color(blue)(-8),color(blue)(+0),color(blue)(+16),,"row 1"), (ul(color(white)("xx")+),,ul(color(white)("xxx")),ul(-2),ul(+4),ul(+8),ul(-16),,"row 2"), (xxcolor(magenta)(""(-2)),,color(red)(1),color(red)(-2),color(red)(-4),color(red)(+8),color(white)("xx")color(green)0,,"row 4"), (,,color(grey)(x^3),color(grey)(x^2),color(grey)(x^1),color(grey)(x^0),color(grey)("Rem."),,) :}

The values for each column for row 4 are the sum of the values in rows 2 and 3 for that column.

The values for each column of row 3 are the product of color(magenta)(""(-2)) and the value in row 4 of the previous column.

Dec 29, 2017

color(magenta)(x^3-2x^2-4x+8 (using long division)

#### Explanation:

$\left({x}^{4} - 8 {x}^{2} + 16\right) \div \left(x + 2\right)$

color(white)(..........)color(white)(.)color(magenta)(x^3-2x^2-4x+8
$x + 2 | \overline{{x}^{4} + 0 {x}^{3} - 8 {x}^{2} + 0 x - 16}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots} \underline{{x}^{4} + 2 {x}^{3}}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots . .} - 2 {x}^{3} - 8 {x}^{2}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots .} \underline{- 2 {x}^{3} - 4 {x}^{2}}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} - 4 {x}^{2} + 0 x$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{- 4 {x}^{2} - 8 x}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} 8 x - 16$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{8 x - 16}$