# How do you divide (-x^4+6x^3+2x^2-7x-7)/(x-2) ?

Apr 25, 2018

$- {x}^{3} + 4 {x}^{2} + 10 x + 13 \text{ rem } 19$

#### Explanation:

This requires long division or the shorter method of synthetic division.

set $x - 2$ equal to $0 \text{ } \rightarrow x = 2$

Write down the coefficients of the $x$ terms in descending order.

$\text{ "2|-1" "6" "2" "-7" } - 7$
$\text{ "|ul(" } \downarrow \textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times x}$
$\textcolor{w h i t e}{\times \times} - 1 \text{ } \leftarrow$ bring down the $- 1$

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$\text{ "color(blue)(2)|-1" "6" "2" "-7" } - 7$
" "|ul(" "darr" "color(blue)(-2)" "larr (color(blue)(2xx-1))" "color(white)(xxxxx)
$\textcolor{w h i t e}{\times \times x} \textcolor{b l u e}{- 1} \text{ "color(red)(4)" } \leftarrow$ add $6 + \left(- 2\right)$

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Repeat the process:

Multiply $2$ outside by the new value at the bottom:

Multiply $\textcolor{red}{2 \times 4 = 8}$ and then add $2 + 8 = \textcolor{\lim e}{10}$

$\text{ "color(red)(2)|-1" "6" "2" "-7" } - 7$
$\text{ "|ul(color(white)(xxxxx)-2" } \textcolor{red}{8} \textcolor{w h i t e}{\times \times \times \times \times \times}$
$\textcolor{w h i t e}{\times \times x} - 1 \text{ "color(red)(4)" } \textcolor{\lim e}{10}$

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Multiply $\textcolor{\lim e}{2 \times 10 = 20}$ and then add $- 7 + 20 = \textcolor{m a \ge n t a}{13}$

$\text{ "color(lime)(2)|-1" "6" "2" "-7" } - 7$
$\text{ "|ul(color(white)(xxxxx)-2" "8 " } \textcolor{\lim e}{20} \textcolor{w h i t e}{\times \times \times \times \times \times}$
$\textcolor{w h i t e}{\times \times x} - 1 \text{ "4" "color(lime)(10)" } \textcolor{m a \ge n t a}{13}$

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Multiply $\textcolor{m a \ge n t a}{2 \times 13 = 26}$ and then add $- 7 + 26 = \textcolor{f \mathmr{and} e s t g r e e n}{19}$

$\text{ "color(magenta)(2)|-1" "6" "2" "-7" } - 7$
" "|ul(color(white)(xxxxx)-2" "8 " "20" "color(magenta)(26) color(white)(xxxxxxxxxxxx)
$\textcolor{w h i t e}{\times \times} - 1 \text{ "4" "10" "color(magenta)(13)" "color(forestgreen)(19)" } \leftarrow$ the remainder

These are the coefficients of the quotient. $\left({x}^{4} \div x = {x}^{3}\right)$

$= - {x}^{3} + 4 {x}^{2} + 10 x + 13 \text{ rem } 19$