# How do you divide (-x^3+13x^2-x-5)/(x-4)?

Apr 29, 2017

$- {x}^{2} + 9 x + 35 + \frac{135}{x - 4}$

#### Explanation:

One way is to use the divisor as a factor in the numerator.

$\textcolor{m a \ge n t a}{\text{add / subtract "" the terms that occur as a consequence}}$

$\text{consider the numerator}$

$\textcolor{red}{- {x}^{2}} \left(x - 4\right) \textcolor{m a \ge n t a}{- 4 {x}^{2}} + 13 {x}^{2} - x - 5$

$= \textcolor{red}{- {x}^{2}} \left(x - 4\right) \textcolor{red}{+ 9 x} \left(x - 4\right) \textcolor{m a \ge n t a}{+ 36 x} - x - 5$

$= \textcolor{red}{- {x}^{2}} \left(x - 4\right) \textcolor{red}{+ 9 x} \left(x - 4\right) \textcolor{red}{+ 35} \left(x - 4\right) \textcolor{m a \ge n t a}{+ 140} - 5$

$\Rightarrow \text{quotient "=color(red)(-x^2+9x+35)" remainder } = 135$

$\Rightarrow \frac{- {x}^{3} + 13 {x}^{2} - x - 5}{x - 4} = - {x}^{2} + 9 x + 35 + \frac{135}{x - 4}$

Apr 29, 2017

$\left(- {x}^{2} + 9 x + 35\right) + 135$

#### Explanation:

For this question, use synthetic division (if you don't know what that is, check this out: http://www.purplemath.com/modules/synthdiv.htm)

x-4 is linear, so we are allowed to use synthetic division.

Your synthetic division should look something like this:

4| -1 13 -1 -5

$$     -4   36  140

-1   9   35   135


So we get that $\frac{- {x}^{3} + 13 {x}^{2} - x - 5}{x - 4} = \left(- {x}^{2} + 9 x + 35\right) + 135$

Alternatively, you can use the long division to do this, though it will take a bit longer (https://www.mathsisfun.com/algebra/polynomials-division-long.html)

Hope that helps!