How do you use synthetic division to divide x^4+4x^3+6x^2+4x+1 by x+1?

Jul 21, 2018

The remainder is $0$ and the quotient is $= {x}^{3} + 3 {x}^{2} + 3 x + 1$

Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$- 1$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$4$$\textcolor{w h i t e}{a a a a a a}$$6$$\textcolor{w h i t e}{a a a a}$$4$$\textcolor{w h i t e}{a a a a a}$$1$

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a}$$- 1$$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{a a a}$$- 3$$\textcolor{w h i t e}{a a a}$$- 1$

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

$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a a a}$$3$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{0}$

The remainder is $0$ and the quotient is $= {x}^{3} + 3 {x}^{2} + 3 x + 1$

$\frac{{x}^{4} + 4 {x}^{3} + 6 {x}^{2} + 4 x + 1}{x + 1} = {x}^{3} + 3 {x}^{2} + 3 x + 1$