How do you use synthetic division to divide ( 2x^3 + 3x^2 + 4 ) / ( 2x + 1 )?

Jul 11, 2018

The remainder is $= \left(\frac{9}{2}\right)$ and the quotient is $= \left({x}^{2} + x - \frac{1}{2}\right)$

Explanation:

Let's perform the synthetic division

Divide by $2$

$\textcolor{w h i t e}{a a a a}$$- \frac{1}{2}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$+ \frac{3}{2}$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$2$

$\textcolor{w h i t e}{a a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$- \frac{1}{2}$$\textcolor{w h i t e}{a a a a a}$$- \frac{1}{2}$$\textcolor{w h i t e}{a a a a}$$\frac{1}{4}$

$\textcolor{w h i t e}{a a a a a a a a a}$_________________________________________________________

$\textcolor{w h i t e}{a a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a a a}$$1$$\textcolor{w h i t e}{a a a a a a}$$- \frac{1}{2}$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{\frac{9}{4}}$

The remainder is $= \left(\frac{9}{2}\right)$ and the quotient is $= \left({x}^{2} + x - \frac{1}{2}\right)$

$\frac{2 {x}^{3} + 3 {x}^{2} + 4}{2 x + 1} = \left({x}^{2} + x - \frac{1}{2}\right) + \frac{\frac{9}{4}}{x + \frac{1}{2}}$

$\frac{2 {x}^{3} + 3 {x}^{2} + 4}{2 x + 1} = \left({x}^{2} + x - \frac{1}{2}\right) + \frac{\frac{9}{2}}{2 x + 1}$