# How do you use synthetic division to show that x=2 is a zero of x^3-7x+6=0?

Jan 25, 2017

#### Explanation:

To divide ${x}^{3} - 7 x + 6 = {x}^{3} + 0 {x}^{2} - 7 x + 6$ by $x - 2$ (as we have to show $x = 2$, a zero).

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

$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)0color(white)(XX)-7" "" } 6$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Two As $x = 2$ we put $2$ at the left.

$2 | \textcolor{w h i t e}{x} 1 \text{ "color(white)(X)0color(white)(XX)-7" "" } 6$
$\textcolor{w h i t e}{x} | \text{ } \textcolor{w h i t e}{X X}$
" "stackrel("—————————————)

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

$2 | \textcolor{w h i t e}{x} 1 \text{ "color(white)(X)0color(white)(XX)-7" "" } 6$
$\textcolor{w h i t e}{x} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{x} | \textcolor{w h i t e}{x} \textcolor{red}{1}$

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

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

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

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

$2 | \textcolor{w h i t e}{x} 1 \text{ "color(white)(X)0color(white)(XX)-7" "" } 6$
$\textcolor{w h i t e}{x} | \text{ } \textcolor{w h i t e}{X X} 2 \textcolor{w h i t e}{\times \times x} 4 \textcolor{w h i t e}{X} - 6$
" "stackrel("—————————————)
$\textcolor{w h i t e}{x} | \textcolor{w h i t e}{x} \textcolor{b l u e}{1} \textcolor{w h i t e}{X 11} \textcolor{red}{2} \textcolor{w h i t e}{X X} \textcolor{red}{-} 3 \textcolor{w h i t e}{X X X} \textcolor{red}{0}$

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

As remainder is $0$, $x = 2$ is a zero of ${x}^{3} - 7 x + 6$.