How do you use synthetic substitution to find x=-3 for g(x)=x^3-8x+6?

Oct 23, 2016

Using synethetic substitution, g(-3)=3.

Explanation:

Synethetic substitution is the process of using synthetic division to find $g \left(x\right)$. To find $g \left(- 3\right)$, use synthetic division with a divisor of $- 3$. The remainder equals $g \left(- 3\right)$

Use the coefficients of the polynomial as the dividend. Note that the ${x}^{2}$ term has a coefficient of zero.

$g \left(x\right) = \textcolor{red}{1} {x}^{3} + \textcolor{red}{0} {x}^{2} \textcolor{red}{- 8} x + \textcolor{red}{6}$

$- 3 | \textcolor{red}{1} \textcolor{w h i t e}{a a a a} \textcolor{red}{0} \textcolor{w h i t e}{a a} \textcolor{red}{- 8} \textcolor{w h i t e}{a a a} \textcolor{red}{6}$
$\textcolor{w h i t e}{a a {a}^{2}} \downarrow \textcolor{w h i t e}{a a a a a a a a a a a a a a a a}$Pull down the $\textcolor{red}{1}$
$\textcolor{w h i t e}{a a {a}^{2} a} \textcolor{m a \ge n t a}{1}$

$- 3 | \textcolor{red}{1} \textcolor{w h i t e}{a a a a} \textcolor{red}{0} \textcolor{w h i t e}{a a} \textcolor{red}{- 8} \textcolor{w h i t e}{a a a} \textcolor{red}{6}$
$\textcolor{w h i t e}{a a {a}^{2}} \downarrow \textcolor{w h i t e}{{a}^{2}} \textcolor{b l u e}{- 3} \textcolor{w h i t e}{a a a a a a a a a a}$ Multiply $- 3 \cdot \textcolor{m a \ge n t a}{1} = \textcolor{b l u e}{- 3}$
color(white)(aaaa^2color(magenta)1color(white)(aa)color(magenta)(-3)color(white)(aaaaaaaaaaa)Add $\textcolor{red}{0} + \textcolor{b l u e}{- 3} = \textcolor{m a \ge n t a}{- 3}$

Repeat the process of multiplying by $- 3$ and adding to the coefficient.

$- 3 | \textcolor{red}{1} \textcolor{w h i t e}{a a a a} \textcolor{red}{0} \textcolor{w h i t e}{a a} \textcolor{red}{- 8} \textcolor{w h i t e}{a a a} \textcolor{red}{6}$
$\textcolor{w h i t e}{a a {a}^{2}} \downarrow \textcolor{w h i t e}{{a}^{2}} \textcolor{b l u e}{- 3} \textcolor{w h i t e}{a a a} \textcolor{b l u e}{9} \textcolor{w h i t e}{a a} \textcolor{b l u e}{- 3}$
color(white)(aaaa^2color(magenta)1color(white)(aa)color(magenta)(-3)color(white)(aaa)color(magenta)1color(white)(aa)aacolor(orange)3

The remainder $\textcolor{\mathmr{and} a n \ge}{3} = g \left(- 3\right)$