# How do you evaluate f(x)=x^3+3x^2-2x+5 at x=-3 using direct substitution and synthetic division?

Jan 31, 2018

Quotient $\textcolor{b l u e}{Q = {x}^{2} - 2}$, Remainder $\textcolor{b l u e}{R = \frac{11}{x + 3}}$

#### Explanation:

Given equation is

$f \left(x\right) = {x}^{3} + 3 {x}^{2} - 2 x + 5$ at x = -3

Substitution

$f \left(x\right) = f \left(3\right) = - {3}^{3} + \left(3 \cdot - {3}^{2}\right) - \left(2 \cdot - 3\right) + 5 = 53$

Synthetic Division

$\textcolor{w h i t e}{a a} - 3 \textcolor{w h i t e}{a a} | \textcolor{w h i t e}{a a} 1 \textcolor{w h i t e}{a a a a} 3 \textcolor{w h i t e}{a a} - 2 \textcolor{w h i t e}{a a} 5$
$\textcolor{w h i t e}{a a a a a a a} | \textcolor{w h i t e}{a a} \downarrow \textcolor{w h i t e}{a} - 3 \textcolor{w h i t e}{a a a a} 0 \textcolor{w h i t e}{a a} 6$
$\textcolor{w h i t e}{a a a a a a a} - - - - - - - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a} 1 \textcolor{w h i t e}{a a a} 0 \textcolor{w h i t e}{a a} - 2 \textcolor{w h i t e}{a a} 11$

Quotient $Q = {x}^{2} - 2$, Remainder $R = \frac{11}{x + 3}$