# How do you solve (x^ { 3} - 4x ^ { 2} - 11x + 30 )\div (x - 2)?

Jan 29, 2018

$\textcolor{g r e e n}{{x}^{2} - 2 x - 15}$

#### Explanation:

$\textcolor{w h i t e}{a a a a} 2 \textcolor{w h i t e}{a a} | \textcolor{w h i t e}{a a} 1 \textcolor{w h i t e}{a a} - 4 \textcolor{w h i t e}{a a} - 11 \textcolor{w h i t e}{a a} 30$

$\textcolor{w h i t e}{a a a a a a a a a a} \downarrow \textcolor{w h i t e}{a a a a} 2 \textcolor{w h i t e}{a a} - 4 \textcolor{w h i t e}{} - 30$

$\textcolor{w h i t e}{a a a a a a a} - - - - - - - - -$

$\textcolor{w h i t e}{a a a a a a a a a a a} 1 \textcolor{w h i t e}{a a} - 2 \textcolor{w h i t e}{a a} - 15 \textcolor{w h i t e}{a a} 0$

${x}^{2} - 2 x - 15$

Jan 29, 2018

${x}^{2} - 2 x - 15$

#### Explanation:

$\text{one way is to use the divisor as a factor in the numerator}$

$\text{consider the numerator}$

$\textcolor{red}{{x}^{2}} \left(x - 2\right) \textcolor{m a \ge n t a}{+ 2 {x}^{2}} - 4 {x}^{2} - 11 x + 30$

$= \textcolor{red}{{x}^{2}} \left(x - 2\right) \textcolor{red}{- 2 x} \left(x - 2\right) \textcolor{m a \ge n t a}{- 4 x} - 11 x + 30$

$= \textcolor{red}{{x}^{2}} \left(x - 2\right) \textcolor{red}{- 2 x} \left(x - 2\right) \textcolor{red}{- 15} \left(x - 2\right) \textcolor{m a \ge n t a}{- 30} + 30$

$= \textcolor{red}{{x}^{2}} \left(x - 2\right) \textcolor{red}{- 2 x} \left(x - 2\right) \textcolor{red}{- 15} \left(x - 2\right) + 0$

$\text{quotient "=color(red)(x^2-2x-15)," remainder} = 0$

$\Rightarrow \frac{{x}^{3} - 4 {x}^{2} - 11 x + 30}{x - 2} = {x}^{2} - 2 x - 15$