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

Dec 3, 2015

$\frac{2 {x}^{3} - 3 {x}^{2} + 3 x - 4}{x - 2} = 2 {x}^{2} + x + 5 + \frac{6}{x - 2}$

#### Explanation:

I know that in some countries, the long division of polynoms is being written in a different way. I will use the one that is familiar for me though and hope that you can convert it into your notation easily. :-)

Let me explain to you how to do the long division - and if you already know, you can skip to the end of the answer to see the whole division there.

========================================

1) First, check that the the terms in your numerator and denominator are ordered by the power of $x$ - this is already the case for you.

2) Now, take the first term - the one with the biggest power - from the numerator and divide it by the first term - the one with the biggest power - from the denominator.

In your case, it's $2 {x}^{3} \div x = 2 {x}^{2}$, so you get:

$\textcolor{w h i t e}{\xi i} \left(\textcolor{red}{2 {x}^{3}} - 3 {x}^{2} \textcolor{w h i t e}{x} + 3 x - 4\right) \div \left(\textcolor{red}{x} - 2\right) = \textcolor{red}{2 {x}^{2}}$

3) As next, you need to backwards multiply your new result ($2 {x}^{2}$) with the denominator, so compute $2 {x}^{2} \cdot \left(x - 2\right) = 2 {x}^{3} - 4 {x}^{2}$ and subtract it from your numerator:

$\textcolor{w h i t e}{\xi i} \left(2 {x}^{3} - 3 {x}^{2} \textcolor{w h i t e}{x} + \textcolor{g r e y}{3 x} - 4\right) \div \left(\textcolor{b l u e}{x - 2}\right) = \textcolor{b l u e}{2 {x}^{2}}$
$- \left(\textcolor{b l u e}{2 {x}^{3} - 4 {x}^{2}}\right)$
$\textcolor{w h i t e}{x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times x} {x}^{2} + \textcolor{g r e y}{3 x}$

4) As expected, you don't have a ${x}^{3}$ term anymore, and your new term with the highest power of $x$ in the numerator is ${x}^{2}$, so you need to divide ${x}^{2}$ by $x$:

$\textcolor{w h i t e}{\xi i} \left(2 {x}^{3} - 3 {x}^{2} \textcolor{w h i t e}{x} + 3 x - 4\right) \div \left(\textcolor{red}{x} - 2\right) = 2 {x}^{2} + \textcolor{red}{x}$
$- \left(2 {x}^{3} - 4 {x}^{2}\right)$
$\textcolor{w h i t e}{x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times x} \textcolor{red}{{x}^{2}} + 3 x$

5) Again, do the backward multiplication and subtract the result:

$\textcolor{w h i t e}{\xi i} \left(2 {x}^{3} - 3 {x}^{2} \textcolor{w h i t e}{x} + 3 x \textcolor{w h i t e}{x} \textcolor{g r e y}{- 4}\right) \div \left(\textcolor{b l u e}{x - 2}\right) = 2 {x}^{2} + \textcolor{b l u e}{x}$
$- \left(2 {x}^{3} - 4 {x}^{2}\right)$
$\textcolor{w h i t e}{x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times x} {x}^{2} + 3 x$
$\textcolor{w h i t e}{\times \xi i x} - \left(\textcolor{b l u e}{{x}^{2} - 2 x}\right)$
$\textcolor{w h i t e}{\times \times \times} \frac{\textcolor{w h i t e}{\times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times \times \times x} 5 x \textcolor{w h i t e}{x} \textcolor{g r e y}{- 4}$

6) As next, divide $5 x$ by $x$...

$\textcolor{w h i t e}{\xi i} \left(2 {x}^{3} - 3 {x}^{2} \textcolor{w h i t e}{x} + 3 x \textcolor{w h i t e}{x} - 4\right) \div \left(\textcolor{red}{x} - 2\right) = 2 {x}^{2} + x + \textcolor{red}{5}$
$- \left(2 {x}^{3} - 4 {x}^{2}\right)$
$\textcolor{w h i t e}{x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times x} {x}^{2} + 3 x$
$\textcolor{w h i t e}{\times \xi i x} - \left({x}^{2} - 2 x\right)$
$\textcolor{w h i t e}{\times \times \times} \frac{\textcolor{w h i t e}{\times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times \times \times x} \textcolor{red}{5 x} \textcolor{w h i t e}{x} - 4$

7)... and perform the backward multiplication and subtraction...

$\textcolor{w h i t e}{\xi i} \left(2 {x}^{3} - 3 {x}^{2} \textcolor{w h i t e}{x} + 3 x \textcolor{w h i t e}{x} - 4\right) \div \left(\textcolor{b l u e}{x - 2}\right) = 2 {x}^{2} + x + \textcolor{b l u e}{5}$
$- \left(2 {x}^{3} - 4 {x}^{2}\right)$
$\textcolor{w h i t e}{x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times x} {x}^{2} + 3 x$
$\textcolor{w h i t e}{\times \xi i x} - \left({x}^{2} - 2 x\right)$
$\textcolor{w h i t e}{\times \times \times} \frac{\textcolor{w h i t e}{\times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times \times \times x} 5 x \textcolor{w h i t e}{x} - 4$
$\textcolor{w h i t e}{\times \times \times \times i} - \left(\textcolor{b l u e}{5 x \textcolor{w h i t e}{x} - 10}\right)$
$\textcolor{w h i t e}{\times \times \times \times \times x} \frac{\textcolor{w h i t e}{\times \times \times x}}{}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times i} 6$

8) At this point, as you can't divide $6 \div x$ anymore, you have finished your division and have the remainder $6$ at the end:

========================================

Solution:

$\frac{2 {x}^{3} - 3 {x}^{2} + 3 x - 4}{x - 2} = 2 {x}^{2} + x + 5 + \frac{6}{x - 2}$