How do you use polynomial synthetic division to divide (2x^3+x^2+2x+1)div(x+1/2) and write the polynomial in the form p(x)=d(x)q(x)+r(x)?

Aug 7, 2017

The answer is $= \left(2 {x}^{2} + 2\right) \left(x + \frac{1}{2}\right)$

Explanation:

Let's perform the synthetic division

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

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a}$$- 1$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$- 1$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$- - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$2$$\textcolor{w h i t e}{a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$2$$\textcolor{w h i t e}{a a a a a a}$$\textcolor{red}{0}$

The remainder is $\textcolor{red}{0}$ and the quotient is $= 2 {x}^{2} + 2$

Therefore,

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