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

Oct 25, 2016

$f \left(- 2\right) = 4$

Explanation:

$f \left(x\right) = 2 {x}^{3} + 5 {x}^{2} + 4 x + 8$ at $x = - 2$

By direct substitution:

$f \left(- 2\right) = 2 {\left(- 2\right)}^{3} + 5 {\left(- 2\right)}^{2} + 4 \left(- 2\right) + 8$

$f \left(- 2\right) = 2 \left(- 8\right) + 5 \left(4\right) - 8 + 8$

$f \left(- 2\right) = \textcolor{red}{4}$

$- 2 | 2 \textcolor{w h i t e}{a a a a} 5 \textcolor{w h i t e}{a a a a a} 4 \textcolor{w h i t e}{a a a} 8$
$\textcolor{w h i t e}{{a}^{2} a a} \downarrow \textcolor{w h i t e}{a} - 4 \textcolor{w h i t e}{a a} - 2 \textcolor{w h i t e}{a} - 4$
$\textcolor{w h i t e}{a a a a} - - - - - - -$
$\textcolor{w h i t e}{a a {a}^{22}} 2 \textcolor{w h i t e}{{a}^{22} a a} 1 \textcolor{w h i t e}{a a a {a}^{2}} 2 \textcolor{w h i t e}{a a a} \textcolor{red}{4}$