How do you evaluate f(x)=x+1/2x^3 at x=4 using direct substitution and synthetic division?

Jan 30, 2018

Direct Substitution : f(x) = f(4) = 36#

Synthetic Division :
Quotient $\left(\frac{1}{2}\right) {x}^{2} + 2 x + 9$, Remainder $= \frac{18}{x - 4}$

Explanation:

For x = 4,

$f \left(x\right) = f \left(4\right) = \left(\frac{1}{2}\right) {x}^{3} + x = \left(\frac{1}{2}\right) {4}^{4} + 4 = 36$

Using Synthetic Division,

$\textcolor{w h i t e}{a a} 4 \textcolor{w h i t e}{a a} | \textcolor{w h i t e}{a a} \frac{1}{2} \textcolor{w h i t e}{a a} 0 \textcolor{w h i t e}{a a} 1 \textcolor{w h i t e}{a a} 0$
$\textcolor{w h i t e}{a a a a a} | \textcolor{w h i t e}{a a} \downarrow \textcolor{w h i t e}{a} 2 \textcolor{w h i t e}{a a} 8 \textcolor{w h i t e}{a a} 18$
$\textcolor{w h i t e}{a a a a a a} - - - - - - -$
$\textcolor{w h i t e}{a a a a a a a a a} \frac{1}{2} \textcolor{w h i t e}{a a} 2 \textcolor{w h i t e}{a a} 9 \textcolor{w h i t e}{a a} 18$

Quotient $\left(\frac{1}{2}\right) {x}^{2} + 2 x + 9$, Remainder $= \frac{18}{x - 4}$