# How do you divide (2x^3-5x^2+4x+12)/(x-7) ?

Jun 16, 2017

$2 {x}^{2} + 9 x + 67 + \frac{481}{x - 7}$

#### Explanation:

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

$\text{consider the numerator}$

$\textcolor{red}{2 {x}^{2}} \left(x - 7\right) \textcolor{m a \ge n t a}{+ 14 {x}^{2}} - 5 {x}^{2} + 4 x + 12$

$= \textcolor{red}{2 {x}^{2}} \left(x - 7\right) \textcolor{red}{+ 9 x} \left(x - 7\right) \textcolor{m a \ge n t a}{+ 63 x} + 4 x + 12$

$= \textcolor{red}{2 {x}^{2}} \left(x - 7\right) \textcolor{red}{+ 9 x} \left(x - 7\right) \textcolor{red}{+ 67} \left(x - 7\right) \textcolor{m a \ge n t a}{+ 469} + 12$

$= \textcolor{red}{2 {x}^{2}} \left(x - 7\right) \textcolor{red}{+ 9 x} \left(x - 7\right) \textcolor{red}{+ 67} \left(x - 7\right) + 481$

$\text{quotient "=color(red)(2x^2+9x+67)", remainder } = 481$

$\Rightarrow \frac{2 {x}^{3} - 5 {x}^{2} + 4 x + 12}{x - 7}$

$= 2 {x}^{2} + 9 x + 67 + \frac{481}{x - 7}$

Jun 16, 2017

color(blue)(2x^2+9x+67 plus remainder of $\textcolor{b l u e}{481}$

#### Explanation:

 color(white)(.............)ul(color(blue)(2x^2+9x+67)
$\textcolor{w h i t e}{a a} x - 7$$|$$2 {x}^{3} - 5 {x}^{2} + 4 x + 12$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots . .} \underline{2 {x}^{3} - 14 {x}^{3}}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} 9 {x}^{2} + 4 x$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{9 {x}^{2} - 63 x}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} 67 x + 12$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{67 x - 469}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 481$