# How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems x^3 - 2x^2 + 5x - 6 divided by x - 3?

Dec 15, 2016

The quotient is $= \left({x}^{2} + x + 8\right)$ and the remainder is $= 18$

#### Explanation:

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

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

Let $x = c$, then $f \left(c\right) = r$

Here, $f \left(x\right) = {x}^{3} - 2 {x}^{2} + 5 x - 6$ and $c = 3$

Therefore,

$f \left(3\right) = {3}^{3} - 2 \cdot {3}^{2} + 5 \cdot 3 - 6 = 27 - 18 + 15 - 6 = 18$

The remainder is $= 18$

Let's do the long division

$\textcolor{w h i t e}{a a a a}$${x}^{3} - 2 {x}^{2} + 5 x - 6$$\textcolor{w h i t e}{a a a a}$∣$x - 3$

$\textcolor{w h i t e}{a a a a}$${x}^{3} - 3 {x}^{2}$$\textcolor{w h i t e}{a a a a a a a a a a a a}$∣${x}^{2} + x + 8$

$\textcolor{w h i t e}{a a a a a}$$0 + {x}^{2} + 5 x$

$\textcolor{w h i t e}{a a a a a a a a}$${x}^{2} - 3 x$

$\textcolor{w h i t e}{a a a a a a a a a}$$0 + 8 x - 6$

$\textcolor{w h i t e}{a a a a a a a a a a a}$$+ 8 x - 24$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a}$$0 + 18$

The quotient is $= \left({x}^{2} + x + 8\right)$ and the remainder is $= 18$