# How do you use synthetic division to divide (x^3-729)/(x-9)?

Jan 23, 2017

Quotient is ${x}^{2} + 9 x + 81$ and remainder is $0$.

#### Explanation:

To divide ${x}^{3} - 729$ or ${x}^{3} + 0 {x}^{2} + 0 x - 729$ by $x + 2$

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

$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)0color(white)(XX)0" "" } - 729$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Two As $x - 9 = 0$ gives $x = 9$ put $9$ at the left.

$9 \textcolor{w h i t e}{X} | 1 \text{ "color(white)(X)0color(white)(XX)0" "" } - 729$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X X}$
" "stackrel("—————————————)

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

$9 \textcolor{w h i t e}{X} | 1 \text{ "color(white)(X)0color(white)(XX)0" "" } - 729$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{} \textcolor{red}{1}$

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

$9 \textcolor{w h i t e}{X} | 1 \text{ "color(white)(X)0color(white)(XX)0" "" } - 729$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X x} 9$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{} \textcolor{b l u e}{1}$

$9 \textcolor{w h i t e}{X} | 1 \text{ "color(white)(X)0color(white)(XX)0" "" } - 729$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X x} 9$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{} \textcolor{b l u e}{1} \textcolor{w h i t e}{\times x} \textcolor{red}{9}$

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

$9 \textcolor{w h i t e}{X} | 1 \text{ "color(white)(X)0color(white)(XX)0" "" } - 729$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X x} 9 \textcolor{w h i t e}{X X} 81 \textcolor{w h i t e}{X X \times} 729$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{} \textcolor{b l u e}{1} \textcolor{w h i t e}{\times x} \textcolor{red}{9} \textcolor{w h i t e}{X X} \textcolor{red}{81} \textcolor{w h i t e}{X X X X X} \textcolor{red}{0}$

Hence, Quotient is ${x}^{2} + 9 x + 81$ and remainder is $0$.