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

Nov 24, 2017

$\frac{4}{3} {x}^{2} + \frac{1}{9} x - \frac{32}{27} \text{ Remainder: } \frac{- 196}{27 \left(3 x - 4\right)}$

#### Explanation:

This is best solved using long division. The question is, what do I have to multiply by $3 x - 4$ to get $4 {x}^{3} - 5 {x}^{2} - 4 x - 12$?

$\text{ "4/3 x^2 + 1/9x - 32/27 " Remainder: } \frac{- 196}{27 \left(3 x - 4\right)}$
$3 x - 4 | \overline{4 {x}^{3} - 5 {x}^{2} - 4 x - 12}$
$\text{ "-ul((4x^3-16/3 x^2)) downarrow " } \downarrow$
$\text{ } \frac{1}{3} {x}^{2} - 4 x$
$\text{ } - \underline{\left(\frac{1}{3} {x}^{2} - \frac{4}{9} x\right)}$
$\text{ } - \frac{32}{9} x - 12$
$\text{ } - \underline{\left(- \frac{32}{9} x + \frac{128}{27}\right)}$
$\text{ } - \frac{196}{27}$

Nov 24, 2017

color(magenta)(1.3x^2+0.1x-1.2 and remainder of color(magenta)(-6

#### Explanation:

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

 color(white)(................)color(magenta)(1.3x^2+0.1x-1.2
$\textcolor{w h i t e}{a} 3 x - 4$$|$$\overline{4 {x}^{3} - 5 {x}^{2} - 4 x - 12}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots . .} \underline{4 {x}^{3} - 5.3 {x}^{2}}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} 0.3 {x}^{2} - 4 x$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{0.3 {x}^{2} - 0.4 x}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} - 3.6 x - 12$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . .} \underline{- 3.6 x + 4.8}$
color(white)(...........................................)color(magenta)(-6

color(magenta)((4x^3-5x^2-4x-12)/(3x-4)=1.3x^2+0.1x-1.2and remainder of color(magenta)(-6