How do you divide (p^5+5p^3-11p^2-25p+29)div(p+6) using synthetic division?

Jun 25, 2017

The remainder is $= - 9073$ and the quotient is $= {p}^{4} - 6 {p}^{3} + 41 {p}^{2} - 257 p + 1517$

Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$- 6$$\textcolor{w h i t e}{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}$$0$$\textcolor{w h i t e}{a a a a a}$$5$$\textcolor{w h i t e}{a a a a}$$- 11$$\textcolor{w h i t e}{a a a a}$$- 25$$\textcolor{w h i t e}{a a a a}$$29$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$_________

$\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}$$- 6$$\textcolor{w h i t e}{a a a a}$$36$$\textcolor{w h i t e}{a a a}$$- 246$$\textcolor{w h i t e}{a a a a}$$1542$$\textcolor{w h i t e}{a a}$$- 9102$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a}$$- 6$$\textcolor{w h i t e}{a a a a}$$41$$\textcolor{w h i t e}{a a a}$$- 257$$\textcolor{w h i t e}{a a a a}$$1517$$\textcolor{w h i t e}{a a}$$\textcolor{red}{- 9073}$

The remainder is $= - 9073$ and the quotient is $= {p}^{4} - 6 {p}^{3} + 41 {p}^{2} - 257 p + 1517$

$\frac{{p}^{5} + 5 {p}^{3} - 11 {p}^{2} - 25 p + 29}{p + 6} = {p}^{4} - 6 {p}^{3} + 41 {p}^{2} - 257 p + 1517 - \frac{9073}{p + 6}$