# How do you use synthetic division to divide (x^4 + 2x^3 - 2x - 1) ÷ (x^3 + 3x^2 + 3x + 1)?

Oct 7, 2015

$x - 1$

#### Explanation:

First, write down the coefficients of each term of the dividend (don't forget $0 {x}^{2}$):

$\textcolor{w h i t e}{X X} 1 \textcolor{w h i t e}{X X} 2 \textcolor{w h i t e}{X X} 0 \textcolor{w h i t e}{X} - 2 \textcolor{w h i t e}{X} - 1$

On the left, write down the negative coefficients of each term in the divisor excluding the first term:

$\textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 1 | \textcolor{w h i t e}{X X} 1 \textcolor{w h i t e}{X X} 2 \textcolor{w h i t e}{X X} 0 \textcolor{w h i t e}{X} - 2 \textcolor{w h i t e}{X} - 1 \textcolor{w h i t e}{X}$

The solution will look like this:

$\textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 1 | \textcolor{w h i t e}{X X} 1 \textcolor{w h i t e}{X X} 2 \textcolor{w h i t e}{X X} 0 \textcolor{w h i t e}{X} - 2 \textcolor{w h i t e}{X} - 1 \textcolor{w h i t e}{X}$
$\textcolor{w h i t e}{\textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 1} | \textcolor{w h i t e}{X X} \textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X X} 3 \textcolor{w h i t e}{X X} 0 \textcolor{w h i t e}{X X X} 0 \textcolor{w h i t e}{X}$
$\textcolor{w h i t e}{\textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 1} | \textcolor{w h i t e}{X X} \textcolor{w h i t e}{X X X} \textcolor{w h i t e}{x} - 3 \textcolor{w h i t e}{X X} 3 \textcolor{w h i t e}{X x X} 0 \textcolor{w h i t e}{X}$
$\textcolor{w h i t e}{\textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 1} | \underline{\textcolor{w h i t e}{X X} \textcolor{w h i t e}{X X X} \textcolor{w h i t e}{X X X} \textcolor{w h i t e}{X} - 1 \textcolor{w h i t e}{X X} 1 \textcolor{w h i t e}{X}}$
$\textcolor{w h i t e}{\textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 3 \textcolor{w h i t e}{X} - 1} \textcolor{w h i t e}{X X X} 1 \textcolor{w h i t e}{X} - 1 \textcolor{w h i t e}{X X} 0 \textcolor{w h i t e}{X X} 0 \textcolor{w h i t e}{X X X} 0 \textcolor{w h i t e}{X}$

Therefore, the quotient is $x - 1$.