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

Jan 15, 2016

${x}^{4} + 2 {x}^{3} + 3 {x}^{2} + 7 x + 12$ with the remainder $19$

#### Explanation:

I know that there are different notations for the polynomial long division in different countries. I will use the one that I'm familiar with and hope that it will be easy for you to adapt it to your notation if needed. :-)

$\textcolor{w h i t e}{\xi i} \left({x}^{5} \textcolor{w h i t e}{\times \times x} - {x}^{3} + {x}^{2} - 2 x - 5\right) \div \left(x - 2\right) = {x}^{4} + 2 {x}^{3} + 3 {x}^{2} + 7 x + 12$
$- \left({x}^{5} - 2 {x}^{4}\right)$
$\textcolor{w h i t e}{\times} \frac{\textcolor{w h i t e}{\times \times \times}}{\textcolor{w h i t e}{x}}$
$\textcolor{w h i t e}{\times \times \times} 2 {x}^{4} - {x}^{3}$
$\textcolor{w h i t e}{\times x} - \left(2 {x}^{4} - 4 {x}^{3}\right)$
$\textcolor{w h i t e}{\times \times x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{\textcolor{w h i t e}{x}}$
$\textcolor{w h i t e}{\times \times \times \times \times} 3 {x}^{3} + {x}^{2}$
$\textcolor{w h i t e}{\times \times \times x} - \left(3 {x}^{3} - 6 {x}^{2}\right)$
$\textcolor{w h i t e}{\times \times \times \times x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{\textcolor{w h i t e}{x}}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times} 7 {x}^{2} - 2 x$
$\textcolor{w h i t e}{\times \times \times \times \times x} - \left(7 {x}^{2} - 14 x\right)$
$\textcolor{w h i t e}{\times \times \times \times \times \times x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{\textcolor{w h i t e}{x}}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times} 12 x - 5$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \xi i} - \left(12 x - 24\right)$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times x} \frac{\textcolor{w h i t e}{\times \times \times \times}}{\textcolor{w h i t e}{x}}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times x} 19$

Thus,

$\frac{{x}^{5} - {x}^{3} + {x}^{2} - 2 x - 5}{x - 2} = {x}^{4} + 2 {x}^{3} + 3 {x}^{2} + 7 x + 12$

with the remainder $19$, or:

$\frac{{x}^{5} - {x}^{3} + {x}^{2} - 2 x - 5}{x - 2} = {x}^{4} + 2 {x}^{3} + 3 {x}^{2} + 7 x + 12 + \frac{19}{x - 2}$