# How do you use synthetic substitution to find x=-3 for  f(x)= -x^4+x^3-x+1?

Jan 30, 2018

$f \left(x\right) = \textcolor{b l u e}{f \left(- 3\right) = - 104}$

Quotient $\textcolor{g r e e n}{Q = - {x}^{3} + 4 {x}^{2} - 12 x + 35}$, Remainder $\textcolor{red}{R = \frac{- 104}{x + 3}}$

#### Explanation:

Direct Substitution : $f \left(x\right) = f \left(- 3\right) = - \left(- {3}^{4}\right) + {\left(- 3\right)}^{3} - \left(- 3\right) + 1$

$f \left(- 3\right) = = - 81 - 27 + 3 + 1 = - 104$

$\textcolor{w h i t e}{a a} - 3 \textcolor{w h i t e}{a a} | \textcolor{w h i t e}{a a} - 1 \textcolor{w h i t e}{a a a} 1 \textcolor{w h i t e}{a a a a a} 0 \textcolor{w h i t e}{a a} - 1 \textcolor{w h i t e}{a a a a a} 1$
$\textcolor{w h i t e}{a a a a a a a a} | \textcolor{w h i t e}{a a a} \downarrow \textcolor{w h i t e}{a a a} 3 \textcolor{w h i t e}{a} - 12 \textcolor{w h i t e}{a a a a} 36 \textcolor{w h i t e}{a} - 105$
$\textcolor{w h i t e}{a a a a a a a} - - - - - - - - - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a} - 1 \textcolor{w h i t e}{a a a a} 4 \textcolor{w h i t e}{a} - 12 \textcolor{w h i t e}{a a a a} 35 \textcolor{w h i t e}{a} - 104$

Quotient $\textcolor{g r e e n}{Q = - {x}^{3} + 4 {x}^{2} - 12 x + 35}$, Remainder $\textcolor{red}{R = \frac{- 104}{x + 3}}$