# How do you divide (x^3+5x^2+7x+2)div(x+2) using synthetic division?

Jan 11, 2017

Quotient is ${x}^{2} + 3 x + 1$ and remainder is $0$

#### Explanation:

To divide ${x}^{3} + 5 {x}^{2} + 7 x + 2$ by $x + 2$

One Write the coefficients of $x$ in the dividend inside an upside-down division symbol.

$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)5color(white)(XX)7" "" } 2$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Two As $x + 2 = 0$ gives $x = - 2$ put $- 2$ at the left.

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)5color(white)(XX)7" "" } 2$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X X}$
" "stackrel("—————————————)

Three Drop the first coefficient of the dividend below the division symbol.

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)5color(white)(XX)7" "" } 2$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{red}{1}$

Four Multiply the result by the constant, and put the product in the next column.

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)5color(white)(XX)7" "" } 2$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X x} - 2$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1}$

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)5color(white)(XX)7" "" } 2$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X x} - 2$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1} \textcolor{w h i t e}{X 11} \textcolor{red}{3}$

Six Repeat Steps Four and Five until you can go no farther.

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)5color(white)(XX)7" "" } 2$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X} - 2 \textcolor{w h i t e}{X} - 6 \textcolor{w h i t e}{X} - 2$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1} \textcolor{w h i t e}{X 11} \textcolor{red}{3} \textcolor{w h i t e}{X X} \textcolor{red}{1} \textcolor{w h i t e}{X X X} \textcolor{red}{0}$

Hence, Quotient is ${x}^{2} + 3 x + 1$ and remainder is $0$.