# How do you divide ( x^3 - 7x - 6)/(x+1)?

May 13, 2018

${x}^{2} - x - 6$

#### Explanation:

Given: $\frac{{x}^{3} - 7 x - 6}{x + 1}$

$\textcolor{B l u e}{\text{Different format but still the traditional long division method.}}$

Using place keepers of zero value. Example: $0 {x}^{2}$

$\textcolor{w h i t e}{\text{ddddddddddd}} {x}^{3} + 0 {x}^{2} - 7 x - 6$
$\textcolor{m a \ge n t a}{{x}^{2}} \left(x + 1\right) \to \underline{2 {x}^{3} + {x}^{2} \leftarrow \text{ Subtract}}$
$\textcolor{w h i t e}{\text{ddddddddddd.d}} 0 - {x}^{2} - 7 x - 6$
$\textcolor{m a \ge n t a}{- x} \left(x + 1\right) \to \textcolor{w h i t e}{\text{ddd")ul(-x^2-x larr" Subtract}}$
$\textcolor{w h i t e}{\text{dddddddddddddddd")0 color(white)("d}} - 6 x - 6$
$\textcolor{m a \ge n t a}{- 6} \left(x + 1\right) \to \textcolor{w h i t e}{\text{dddddd.d")ul(-6x-6larr" Subtract}}$
$\textcolor{w h i t e}{\text{ddddddddddddddddddddd}} 0 + 0$

$\textcolor{m a \ge n t a}{{x}^{2} - x - 6}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Synthetic division method.}}$

Consider the denominator of $x + 1 = 0 \implies x = \textcolor{red}{- 1}$

${x}^{3} + 0 {x}^{2} - 7 x - 6$
$\textcolor{g r e e n}{1 \textcolor{w h i t e}{\text{..")+0color(white)("d.")-7color(white)("d}} - 6}$

$\textcolor{w h i t e}{\text{dddd")color(red)(-1) | color(white)("d}} \textcolor{g r e e n}{1 + 0 - 7 - 6}$
color(white)("ddddd..d")ul(|color(white)(".")darr -1+1+6" "
$\textcolor{w h i t e}{\text{ddddddddd")1color(white)("d}} - 1 - 6 + 0$
$\textcolor{w h i t e}{\text{ddddddd.d")darrcolor(white)("dd")darrcolor(white)("d}} \downarrow$
$\textcolor{w h i t e}{\text{dddddddd}} 1 {x}^{2} - 1 x - 6$
$\textcolor{m a \ge n t a}{\textcolor{w h i t e}{\text{ddddddddd")x^2-color(white)("d}} x - 6}$