How do you evaluate f(x)=-3x^3+7x^2-4x+8 at x=3 using direct substitution and synthetic division?

Aug 10, 2017

The remainder is $\textcolor{red}{- 22}$ and the quotient is $= - 3 {x}^{2} - 2 x - 10$

Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$3$$\textcolor{w h i t e}{a a a a a a}$$|$$\textcolor{w h i t e}{a a}$$- 3$$\textcolor{w h i t e}{a a a a a a a}$$7$$\textcolor{w h i t e}{a a a a a a}$$- 4$$\textcolor{w h i t e}{a a a a a a a}$$8$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$- - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a}$$- 9$$\textcolor{w h i t e}{a a a a a a}$$- 6$$\textcolor{w h i t e}{a a a a a}$$- 30$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$$- - - - - - - - - - - -$

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a}$$- 3$$\textcolor{w h i t e}{a a a a a}$$- 2$$\textcolor{w h i t e}{a a a a a}$$- 10$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{- 22}$

The remainder is $\textcolor{red}{- 22}$ and the quotient is $= - 3 {x}^{2} - 2 x - 10$

Therefore,

$\left(- 3 {x}^{3} + 7 {x}^{2} - 4 x + 8\right) = \left(- 3 {x}^{2} - 2 x - 10\right) - \frac{22}{x - 3}$

Also,

$f \left(3\right) = - 3 \cdot {\left(3\right)}^{3} + 7 \cdot {\left(3\right)}^{2} - 4 \cdot \left(3\right) \cdot 8 = - 81 + 63 - 12 + 8 = - 93 + 71 = - 22$