# How do you divide (-3x^3-3x^2-4x+1)/(3x-4) ?

Oct 15, 2017

I prefer long division.

#### Explanation:

$\frac{- 3 {x}^{3} - 3 {x}^{2} - 4 x + 1}{3 x - 4}$

In long division form:

$\textcolor{w h i t e}{\frac{3 x - 4}{\textcolor{b l a c k}{3 x - 4}}} \frac{\textcolor{w h i t e}{\left(- 3 {x}^{3} - 3 {x}^{2} - 4 x + 1\right)}}{\text{) } - 3 {x}^{3} - 3 {x}^{2} - 4 x + 1}$

Write $- {x}^{2}$ in the quotient:

color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2color(white)(-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)

Multiply $- {x}^{2} \left(3 x - 4\right) = - 3 {x}^{2} + 4 {x}^{2}$ and subtract from the dividend:

color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2color(white)(-3x^2-4x+1))/(") "-3x^3-3x^2-4x+1)
$\textcolor{w h i t e}{\text{.....................}} \underline{3 {x}^{2} - 4 {x}^{2}}$
$\textcolor{w h i t e}{\text{.................... .}} - 7 {x}^{2} - 4 x$

Write $- \frac{7}{3} x$ in the quotient:

color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2-7/3xcolor(white)(-4x+1))/(") "-3x^3-3x^2-4x+1)
$\textcolor{w h i t e}{\text{.....................}} \underline{3 {x}^{2} - 4 {x}^{2}}$
$\textcolor{w h i t e}{\text{.................... .}} - 7 {x}^{2} - 4 x$

Multiply $- \frac{7}{3} x \left(3 x - 4\right) = - 7 {x}^{2} + \frac{28}{3} x$ and subtract from the dividend:

color(white)( (3x-4)/color(black)(3x-4)) (color(white)("....")-x^2-7/3xcolor(white)(-4x+1))/(") "-3x^3-3x^2-4x+1)
$\textcolor{w h i t e}{\text{.....................}} \underline{3 {x}^{2} - 4 {x}^{2}}$
$\textcolor{w h i t e}{\text{.................... .}} - 7 {x}^{2} - 4 x$
$\textcolor{w h i t e}{\text{....................... .}} \underline{7 {x}^{2} - \frac{28}{3} x}$
$\textcolor{w h i t e}{\text{............................... .}} - \frac{40}{3} x + 1$

Write $- \frac{40}{9}$ in the quotient:

color(white)( (3x-4)/color(black)(3x-4)) (color(white)("...")-x^2-7/3x-40/9color(white)(1))/(") "-3x^3-3x^2-4x+1)
$\textcolor{w h i t e}{\text{.....................}} \underline{3 {x}^{2} - 4 {x}^{2}}$
$\textcolor{w h i t e}{\text{.................... .}} - 7 {x}^{2} - 4 x$
$\textcolor{w h i t e}{\text{....................... .}} \underline{7 {x}^{2} - \frac{28}{3} x}$
$\textcolor{w h i t e}{\text{............................... .}} - \frac{40}{3} x + 1$

Multiply $- \frac{40}{9} \left(3 x - 4\right) = - \frac{40}{9} x + \frac{160}{9}$ and subtract from the dividend:

color(white)( (3x-4)/color(black)(3x-4)) (color(white)("...")-x^2-7/3x-40/9color(white)(1))/(") "-3x^3-3x^2-4x+1)
$\textcolor{w h i t e}{\text{.....................}} \underline{3 {x}^{2} - 4 {x}^{2}}$
$\textcolor{w h i t e}{\text{.................... .}} - 7 {x}^{2} - 4 x$
$\textcolor{w h i t e}{\text{....................... .}} \underline{7 {x}^{2} - \frac{28}{3} x}$
$\textcolor{w h i t e}{\text{............................... .}} - \frac{40}{3} x + 1$
$\textcolor{w h i t e}{\text{............................... .}} \underline{\frac{40}{9} x - \frac{160}{9}}$
$\textcolor{w h i t e}{\text{........................................... .}} - \frac{151}{9}$

The quotient is $- {x}^{2} - \frac{7}{3} x - \frac{40}{9}$ with a remainder of $- \frac{151}{9}$