# How do you evaluate f(x)=11x^3-6x^2+2 at x=0 using direct substitution and synthetic division?

Jan 30, 2018

$f \left(x\right) = f \left(0\right) = 2$

Synthetic Division
Quotient $\textcolor{b r o w n}{= 11 {x}^{2} - 6 x}$, Remainder $\textcolor{red}{= \frac{2}{x}}$

#### Explanation:

For x = 0, by Direct Substitution

$f \left(x\right) = f \left(0\right) = 11 \cdot 0 - 6 \cdot 0 + 2 = 2$

Synthetic Division

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

$\textcolor{w h i t e}{a a a a a} | \textcolor{w h i t e}{a a} \downarrow \textcolor{w h i t e}{a a a a} 0 \textcolor{w h i t e}{a a a} 0 \textcolor{w h i t e}{a a a} 0$

$\textcolor{w h i t e}{a a a a a} - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a a a a a a} 11 \textcolor{w h i t e}{a a} - 6 \textcolor{w h i t e}{a a a} 0 \textcolor{w h i t e}{a a a} 2$

Quotient $= 11 {x}^{2} - 6 x$, Remainder $= \frac{2}{x}$