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

Jul 21, 2018

The remainder is $- 31$ and the quotient is $= - 4 {x}^{2} - 8 x - 13$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$2$$|$$\textcolor{w h i t e}{a a a a}$$- 4$$\textcolor{w h i t e}{a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a a a}$$- 5$

$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a}$$- 8$$\textcolor{w h i t e}{a a a a}$$- 16$$\textcolor{w h i t e}{a a a a a}$$- 26$

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

$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$- 4$$\textcolor{w h i t e}{a a a a}$$- 8$$\textcolor{w h i t e}{a a a a}$$- 13$$\textcolor{w h i t e}{a a a a a}$$\textcolor{red}{- 31}$

The remainder is $- 31$ and the quotient is $= - 4 {x}^{2} - 8 x - 13$

$\frac{- 4 {x}^{3} + 3 x - 5}{x - 2} = - 4 {x}^{2} - 8 x - 13 - \frac{31}{x - 2}$

Apply the remainder theorem

When a polynomial $f \left(x\right)$ is divided by $\left(x - c\right)$, we get

$f \left(x\right) = \left(x - c\right) q \left(x\right) + r$

Let $x = c$

Then,

$f \left(c\right) = 0 + r$

Here,

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

Therefore,

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

$= - 32 + 6 - 5$

$= - 31$