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

Dec 15, 2015

${x}^{3} - \frac{3}{2} {x}^{2} + \frac{9}{4} x - \frac{11}{8}$ with remainder $\frac{41}{8}$

#### Explanation:

I know that there are in some countries, a different notation of long polynomial division is being used. Let me use the notation that I'm most familiar with, I hope that it will be no problem for you to convert it into your prefered notation.

$\textcolor{w h i t e}{\xi i} \left(2 {x}^{4} \textcolor{w h i t e}{\times \times \times \times \times x} + 4 x + 1\right) \div \left(2 x + 3\right) = {x}^{3} - \frac{3}{2} {x}^{2} + \frac{9}{4} x - \frac{11}{8}$
$- \left(2 {x}^{4} + 3 {x}^{3}\right)$
$\textcolor{w h i t e}{\times} \frac{\textcolor{w h i t e}{\times \times \times x}}{}$
$\textcolor{w h i t e}{\times \times x} - 3 {x}^{3}$
$\textcolor{w h i t e}{\times} - \left(- 3 {x}^{3} - \frac{9}{2} {x}^{2}\right)$
$\textcolor{w h i t e}{\times \times} \frac{\textcolor{w h i t e}{\times \times \times \times \times x}}{}$
$\textcolor{w h i t e}{\times \times \times \times \times \times} \frac{9}{2} {x}^{2} + 4 x$
$\textcolor{w h i t e}{\times \times \times \times x} - \left(\frac{9}{2} {x}^{2} + \frac{27}{4} x\right)$
$\textcolor{w h i t e}{\times \times \times \times \times x} \frac{\textcolor{w h i t e}{\times \times \times \times \times}}{}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times x} - \frac{11}{4} x + \textcolor{w h i t e}{i} 1$
$\textcolor{w h i t e}{\times \times \times \times \times \times} - \left(- \frac{11}{4} x - \frac{33}{8}\right)$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times} \frac{\textcolor{w h i t e}{\times \times \times \times \times x}}{}$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times x} \frac{41}{8}$

${x}^{3} - \frac{3}{2} {x}^{2} + \frac{9}{4} x - \frac{11}{8}$
and your remainder is $\frac{41}{8}$.
$\frac{{x}^{4} + 4 x + 1}{2 x + 3} = {x}^{3} - \frac{3}{2} {x}^{2} + \frac{9}{4} x - \frac{11}{8} + \frac{41}{8 \left(2 x + 3\right)}$