# How do you divide (x^2-5x-5x^3+x^4)div(5+x) using synthetic division?

Dec 5, 2016

The remainder is $= 1300$ and the quotient is $= {x}^{3} - 10 {x}^{2} + 51 x - 260$

#### Explanation:

Rearrange the polynomials in decreasing powers of $x$

Let's do the long division

$\textcolor{w h i t e}{a a a a}$${x}^{4} - 5 {x}^{3} + {x}^{2} - 5 x$$\textcolor{w h i t e}{a a a a}$∣$x + 5$

$\textcolor{w h i t e}{a a a a}$${x}^{4} + 5 {x}^{3}$$\textcolor{w h i t e}{a a a a a a a a a a a a a}$∣${x}^{3} - 10 {x}^{2} + 51 x - 260$

$\textcolor{w h i t e}{a a a a}$$0 - 10 {x}^{3} + {x}^{2}$

$\textcolor{w h i t e}{a a a a a a}$$- 10 {x}^{3} - 50 {x}^{2}$

$\textcolor{w h i t e}{a a a a a a a a a a}$$0 + 51 {x}^{2} - 5 x$

$\textcolor{w h i t e}{a a a a a a a a a a a a}$$+ 51 {x}^{2} + 255 x$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a}$$0 - 260 x$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a}$$- 260 x - 1300$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a}$$0 + 1300$

You can use the remainder theorem

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

$f \left(- 5\right) = 625 + 625 + 25 + 25 = 1300$