# How do you evaluate f(x)=5x^3+4x^2+8x+1 at x=2 using direct substitution and synthetic division?

Jan 30, 2018

$f \left(x\right) = f \left(2\right) = \textcolor{red}{73}$

Synthetic Division :
Quotient $\textcolor{b r o w n}{Q = 5 {x}^{2} + 14 x + 36}$, Remainder color(red)(R = 73 / (x-2)

#### Explanation:

Using direct substitution:

$f \left(x\right) = \textcolor{red}{f \left(2\right)} = 5 \cdot \left({2}^{3}\right) + 4 \cdot \left({2}^{2}\right) + \left(8 \cdot 2\right) + 1 = 40 + 16 + 16 + 1 = \textcolor{red}{73}$

Synthetic Division :

$\textcolor{w h i t e}{a a} 2 \textcolor{w h i t e}{a a} | \textcolor{w h i t e}{a a a} 5 \textcolor{w h i t e}{a a a} 4 \textcolor{w h i t e}{a a a} 8 \textcolor{w h i t e}{a a a} 1$
$\textcolor{w h i t e}{a a a a a a} | \textcolor{w h i t e}{a} \downarrow \textcolor{w h i t e}{a a} 10 \textcolor{w h i t e}{a a} 28 \textcolor{w h i t e}{a a} 72$
$\textcolor{w h i t e}{a a a a a a} - - - - - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a} 5 \textcolor{w h i t e}{a a} 14 \textcolor{w h i t e}{a a} 36 \textcolor{w h i t e}{a a} 73$

Quotient $\textcolor{b r o w n}{Q = 5 {x}^{2} + 14 x + 36}$, Remainder color(red)(R = 73 / (x-2)