# How do you divide 5x^2 - 6x^3 + 1 + 7x by 3x - 4?

Jan 29, 2016

$\left(- 6 {x}^{3} + 5 {x}^{2} + 7 x + 1\right) \div \left(3 x - 4\right) = - 2 {x}^{2} - x + 1$

with remainder $5$.

#### Explanation:

First of all, order the terms by the power of $x$. In your case, this means

$- 6 {x}^{3} + 5 {x}^{2} + 7 x + 1$

for the first term.

Now let me walk you through the polynomial long division.

You are basically doing the following operations:

• Divide the dividend's term with the highest power by the divisor's term with the highest power. In your case, that's $\left(- 6 {x}^{3}\right) \div \left(3 x\right) = - 2 {x}^{2}$

• Multiply the result, in your case $- 2 {x}^{2}$, with the divisor: $\left(- 2 {x}^{2}\right) \cdot \left(3 x - 4\right) = - 6 {x}^{3} + 8 {x}^{2}$

• Subtract the result from the last step from your divident: $\left(- 6 {x}^{3} + 5 {x}^{2} + 7 x + 1\right) - \left(- 6 {x}^{3} + 8 {x}^{2}\right) = - 3 {x}^{2} + 7 x + 1$

• Now, you can repeat all those steps with the term $- 3 {x}^{2} + 7 x + 1$ as a new divident.... etc.

In total, your division should look like this:

$\textcolor{w h i t e}{\times} \left(- 6 {x}^{3} + 5 {x}^{2} + 7 x + 1\right) \div \left(3 x - 4\right) = - 2 {x}^{2} - x + 1$
$- \left(- 6 {x}^{3} + 8 {x}^{2}\right)$
 color(white)(xx) color(white)(xxxxxxxxx) /
$\textcolor{w h i t e}{\times \times \times x} - 3 {x}^{2} + 7 x$
$\textcolor{w h i t e}{\times \times x} - \left(- 3 {x}^{2} + 4 x\right)$
 color(white)(xxxxxxx) color(white)(xxxxxxxxx) /
$\textcolor{w h i t e}{\times \times \times \times \times \times \times} 3 x + 1$
$\textcolor{w h i t e}{\times \times \times \times \times \xi i} - \left(3 x - 4\right)$
 color(white)(xxxxxxxxxxxxx) color(white)(xxxxxxxxx) /
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times} 5$

This means that

$\left(- 6 {x}^{3} + 5 {x}^{2} + 7 x + 1\right) \div \left(3 x - 4\right) = - 2 {x}^{2} - x + 1$

with remainder $5$.

Or, if you prefer a different notation,

$\left(- 6 {x}^{3} + 5 {x}^{2} + 7 x + 1\right) \div \left(3 x - 4\right) = - 2 {x}^{2} - x + 1 + \frac{5}{3 x - 4}$