# How do you divide (4x^2+x+1)div(x-2) using synthetic division?

Oct 16, 2016

The quotient is $4 x + 9 + \frac{19}{x - 2}$.

#### Explanation:

$\left(\textcolor{red}{4} {x}^{2} + \textcolor{red}{1} x + \textcolor{red}{1}\right) \div \left(x - 2\right)$ using synthetic division.

Set up using the zero of the dividend $\left(x - 2\right)$, or $x = \textcolor{b l u e}{2}$, and the coefficients $\textcolor{red}{4 \textcolor{w h i t e}{a} 1 \textcolor{w h i t e}{a} 1}$ of the dividend.

$\textcolor{b l u e}{2} | \textcolor{red}{4} \textcolor{w h i t e}{a a a} \textcolor{red}{1} \textcolor{w h i t e}{a a a} \textcolor{red}{1}$

$\textcolor{b l u e}{2} | \textcolor{red}{4} \textcolor{w h i t e}{a a a} \textcolor{red}{1} \textcolor{w h i t e}{a a a} \textcolor{red}{1}$
$\textcolor{w h i t e}{a a} \downarrow$
$\textcolor{w h i t e}{{a}^{2} a} \textcolor{m a \ge n t a}{4} \textcolor{w h i t e}{a a a a a a a a a a a}$Pull down the 4

$\textcolor{b l u e}{2} | \textcolor{red}{4} \textcolor{w h i t e}{a a a} \textcolor{red}{1} \textcolor{w h i t e}{a a a} \textcolor{red}{1}$
$\textcolor{w h i t e}{a a} \downarrow \textcolor{w h i t e}{{a}^{11}} \textcolor{\lim e g r e e n}{8} \textcolor{w h i t e}{a a a a a a a}$Multiply $\textcolor{b l u e}{2} \times \textcolor{m a \ge n t a}{4} = \textcolor{\lim e g r e e n}{8}$
$\textcolor{w h i t e}{{a}^{2} a} \textcolor{m a \ge n t a}{4} \textcolor{w h i t e}{a a a a a a a a a}$

$\textcolor{b l u e}{2} | \textcolor{red}{4} \textcolor{w h i t e}{a a a} \textcolor{red}{1} \textcolor{w h i t e}{a a a} \textcolor{red}{1}$
$\textcolor{w h i t e}{a a} \downarrow \textcolor{w h i t e}{{a}^{11}} \textcolor{\lim e g r e e n}{8}$
$\textcolor{w h i t e}{{a}^{2} a} \textcolor{m a \ge n t a}{4} \textcolor{w h i t e}{a a a} \textcolor{m a \ge n t a}{9} \textcolor{w h i t e}{a a a a a a a a}$Add $\textcolor{red}{1} + \textcolor{\lim e g r e e n}{8} = \textcolor{m a \ge n t a}{9}$

$\textcolor{b l u e}{2} | \textcolor{red}{4} \textcolor{w h i t e}{a a a} \textcolor{red}{1} \textcolor{w h i t e}{a a a} \textcolor{red}{1}$
$\textcolor{w h i t e}{a a} \downarrow \textcolor{w h i t e}{{a}^{11}} \textcolor{\lim e g r e e n}{8} \textcolor{w h i t e}{a a} \textcolor{\lim e g r e e n}{18} \textcolor{w h i t e}{a a a}$Multiply $\textcolor{b l u e}{2} \times \textcolor{m a \ge n t a}{9} = \textcolor{\lim e g r e e n}{18}$
$\textcolor{w h i t e}{{a}^{2} a} \textcolor{m a \ge n t a}{4} \textcolor{w h i t e}{a a a} \textcolor{m a \ge n t a}{9} \textcolor{w h i t e}{a a a a a a a a}$

$\textcolor{b l u e}{2} | \textcolor{red}{4} \textcolor{w h i t e}{a a a} \textcolor{red}{1} \textcolor{w h i t e}{a a a} \textcolor{red}{1}$
$\textcolor{w h i t e}{a a} \downarrow \textcolor{w h i t e}{{a}^{11}} \textcolor{\lim e g r e e n}{8} \textcolor{w h i t e}{a a} \textcolor{\lim e g r e e n}{18} \textcolor{w h i t e}{a a a}$
$\textcolor{w h i t e}{{a}^{2} a} \textcolor{m a \ge n t a}{4} \textcolor{w h i t e}{a a a} \textcolor{m a \ge n t a}{9} \textcolor{w h i t e}{a a} \textcolor{m a \ge n t a}{19} \textcolor{w h i t e}{a a a a}$Add $\textcolor{red}{1} + \textcolor{\lim e g r e e n}{18} = \textcolor{m a \ge n t a}{19}$

The $\textcolor{m a \ge n t a}{4}$ and the $\textcolor{m a \ge n t a}{9}$ represent the coefficients of the quotient. The $\textcolor{m a \ge n t a}{19}$ is the numerator of the remainder.

The quotient is then $\textcolor{m a \ge n t a}{4} x + \textcolor{m a \ge n t a}{9} + \frac{\textcolor{m a \ge n t a}{19}}{x - 2}$

Oct 16, 2016

The quotient is 4x + 9 + 19/(x - 2.

#### Explanation:

Your divisor, $x - 2$, is in $x - k$ form, so $k = 2$. Your dividend, $4 {x}^{2} + x + 1$, is in standard form, as it needs to be. There are no "missing" terms - the terms have degrees of $2 , 1$, and $0$. So the coefficients which will be used in the synthetic division are (in order):
$4 , 1$, and $1$. Initially, you will set up the synthetic division like this:

2 | 4 1 1

The process is to bring down the first coefficient (4). Then, write the product of this coefficient (4) and the divisor (2) under the next coefficient (1). Add the second coefficient (1) and the product (8), and write this sum (9) in the answer line to the right of the 4. Multiply this sum (9) and the divisor (2) and write their product (18) under the last coefficient (1). Add this product (18) and the last coefficient (1) and write their sum (19) in the answer line to the right of the 9.

This gives you the coefficient answer line 4 9 19. This line means that the quotient of $\left(4 {x}^{2} + x + 1\right)$ and $\left(x - 2\right)$ is:
$4 x + 9 + \frac{19}{x - 2}$.

I can't get this program to line everything up correctly, so I can't show you how to do this very well. Here is my attempt:

' | 4 1 1
2|818
' 4 9 | 19