# How do you divide (5x^3-7x^2-4x+1)/(3x-1) ?

Feb 4, 2018

Use long division.

$\frac{5}{3} {x}^{2} - \frac{16}{9} x - \frac{52}{27} - \frac{25}{27 \left(3 x - 1\right)}$

#### Explanation:

$\left(3 x - 1\right)$ will go into $\left(5 {x}^{3} - 7 {x}^{2} - 4 x + 1\right)$ a total of $\frac{5}{3} {x}^{2}$ times.

$\frac{5}{3} {x}^{2} \left(3 x - 1\right) = \left(5 {x}^{3} - \frac{5}{3} {x}^{2}\right)$

This will leave a remainder of:

$\text{ "" "" "" } \frac{5}{3} {x}^{2}$
$\text{ "" "" "" ""----------------------------}$
$3 x - 1 \text{ } | 5 {x}^{3} - 7 {x}^{2} - 4 x + 1$
$\text{ "" "" } - \left(5 {x}^{3} - \frac{5}{3} {x}^{2}\right)$
$\text{ "" "" "" ""--------------}$
$\text{ "" "" "" "" "" } - \frac{16}{3} {x}^{2}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So now we're left with $- \frac{16}{3} {x}^{2} - 4 x + 1$.

$\left(3 x - 1\right)$ will go into this $- \frac{16}{9} x$ times.

$- \frac{16}{9} x \left(3 x - 1\right) = \left(- \frac{16}{3} {x}^{2} + \frac{16}{9} x\right)$

This will leave a remainder of:

$\text{ "" "" "" } \frac{5}{3} {x}^{2} - \frac{16}{9} x$
$\text{ "" "" "" ""----------------------------}$
$3 x - 1 \text{ } | 5 {x}^{3} - 7 {x}^{2} - 4 x + 1$
$\text{ "" "" } - \left(5 {x}^{3} - \frac{5}{3} {x}^{2}\right)$
$\text{ "" "" "" ""-----------------}$
$\text{ "" "" "" "" "" } - \frac{16}{3} {x}^{2} - 4 x + 1$
$\text{ "" "" "" } - \left(- \frac{16}{3} {x}^{2} + \frac{16}{9} x\right)$
$\text{ "" "" "" "" ""-------------------------}$
$\text{ "" "" "" "" "" "" "" "" } - \frac{52}{9} x$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So now we're left with $- \frac{52}{9} x + 1$.

$\left(3 x - 1\right)$ will go into this $- \frac{52}{27}$ times.

$- \frac{52}{27} \left(3 x - 1\right) = \left(- \frac{52}{9} x + \frac{52}{27}\right)$

This will leave a remainder of:

$\text{ "" "" "" } \frac{5}{3} {x}^{2} - \frac{16}{9} x - \frac{52}{27}$
$\text{ "" "" "" ""----------------------------}$
$3 x - 1 \text{ } | 5 {x}^{3} - 7 {x}^{2} - 4 x + 1$
$\text{ "" "" } - \left(5 {x}^{3} - \frac{5}{3} {x}^{2}\right)$
$\text{ "" "" "" ""-----------------}$
$\text{ "" "" "" "" "" } - \frac{16}{3} {x}^{2} - 4 x + 1$
$\text{ "" "" "" } - \left(- \frac{16}{3} {x}^{2} + \frac{16}{9} x\right)$
$\text{ "" "" "" "" ""-------------------------}$
$\text{ "" "" "" "" "" "" "" "" } - \frac{52}{9} x + 1$
$\text{ "" "" "" "" "" "" } - \left(- \frac{52}{9} x + \frac{52}{27}\right)$
$\text{ "" "" "" "" "" "" ""-------------------------}$
$\textcolor{w h i t e}{\text{MMMMMMMMMMMMMM-}} - \frac{25}{27}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Since the degree of this remainder is smaller than the degree of our divisor, we will just divide it by the divisor as the last term in our answer:

$\frac{5}{3} {x}^{2} - \frac{16}{9} x - \frac{52}{27} - \frac{\frac{25}{27}}{3 x - 1}$

$\frac{5}{3} {x}^{2} - \frac{16}{9} x - \frac{52}{27} - \frac{25}{27 \left(3 x - 1\right)}$