# How do you use polynomial synthetic division to divide (x^6-6x^4+12x^2-8)div(x+sqrt2) and write the polynomial in the form p(x)=d(x)q(x)+r(x)?

Jul 10, 2017

$q \left(x\right) = {x}^{5} - {x}^{4} \sqrt{2} - 4 {x}^{3} + 4 {x}^{2} \sqrt{2} + 20 x - 20 \sqrt{2}$
$r \left(x\right) = 32$

#### Explanation:

Briot Ruffini

$\left(\begin{matrix}\null \\ - \sqrt{2}\end{matrix}\right) \left(\begin{matrix}1 \\ 1\end{matrix}\right) \left(\begin{matrix}0 \\ - \sqrt{2}\end{matrix}\right) \left(\begin{matrix}- 6 \\ - 4\end{matrix}\right) \left(\begin{matrix}0 \\ 4 \sqrt{2}\end{matrix}\right) \left(\begin{matrix}12 \\ 20\end{matrix}\right) \left(\begin{matrix}0 \\ - 20 \sqrt{2}\end{matrix}\right) \left(\begin{matrix}- 8 \\ 32\end{matrix}\right)$

$p \left(x\right) = d \left(x\right) q \left(x\right) + r \left(x\right)$

$d \left(x\right) = x + \sqrt{2}$

$q \left(x\right) = {x}^{5} - {x}^{4} \sqrt{2} - 4 {x}^{3} + 4 {x}^{2} \sqrt{2} + 20 x - 20 \sqrt{2}$

$r \left(x\right) = 32$