# How do you divide (x^3+x+3)/(x-5)?

Oct 28, 2017

The remainder is $\textcolor{red}{133}$ and the quotient is $= {x}^{2} + 5 x + 26$

#### Explanation:

Let's perform a synthetic division

$\textcolor{w h i t e}{a a}$$5$$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$1$$\textcolor{w h i t e}{a a a a a a a a a a}$$3$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$- - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$5$$\textcolor{w h i t e}{a a a a a a}$$25$$\textcolor{w h i t e}{a a a a a a a a}$$130$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$- - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$5$$\textcolor{w h i t e}{a a a a a a}$$26$$\textcolor{w h i t e}{a a a a a a a a}$$\textcolor{red}{133}$

The remainder is $\textcolor{red}{133}$ and the quotient is $= {x}^{2} + 5 x + 26$

$\frac{{x}^{3} + x + 3}{x - 5} = {x}^{2} + 5 x + 26 + \frac{133}{x - 5}$

Oct 28, 2017

${x}^{2} + 5 x + 26 + \frac{133}{x - 5}$

#### Explanation:

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

$\text{consider the numerator}$

$\textcolor{red}{{x}^{2}} \left(x - 5\right) \textcolor{m a \ge n t a}{+ 5 {x}^{2}} + x + 3$

$= \textcolor{red}{{x}^{2}} \left(x - 5\right) \textcolor{red}{+ 5 x} \left(x - 5\right) \textcolor{m a \ge n t a}{+ 25 x} + x + 3$

$= \textcolor{red}{{x}^{2}} \left(x - 5\right) \textcolor{red}{+ 5 x} \left(x - 5\right) \textcolor{red}{+ 26} \left(x - 5\right) \textcolor{m a \ge n t a}{+ 130} + 3$

$= \textcolor{red}{{x}^{2}} \left(x - 5\right) \textcolor{red}{+ 5 x} \left(x - 5\right) \textcolor{red}{+ 26} \left(x - 5\right) + 133$

$\text{quotient "=color(red)(x^2+5x+26)," remainder} = 133$

$\Rightarrow \frac{{x}^{3} + x + 3}{x - 5} = {x}^{2} + 5 x + 26 + \frac{133}{x - 5}$

Oct 28, 2017

Quotient$= {x}^{2} + 5 x + 26$ and remainder$\frac{133}{x - 5}$

#### Explanation:

$\textcolor{w h i t e}{\ldots \ldots \ldots .} \textcolor{w h i t e}{.} {x}^{2} + 5 x + 26$
$x - 5 | \overline{{x}^{3} + 0 + x + 3}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots} \underline{{x}^{3} - 5 {x}^{2}}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 5 {x}^{2} + x$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots .} \underline{5 {x}^{2} - 25 x}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 26 x + 3$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} \underline{26 x - 130}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} 133$

$\frac{{x}^{3} + x + 3}{x - 5} = {x}^{2} + 5 x + 26 + \frac{133}{x - 5}$