# How do you use synthetic division to show that x=1/2 is a zero of 2x^3-15x^2+27x-10=0?

Jul 12, 2018

#### Explanation:

Let's perform the synthetic division

Divide by $\frac{1}{2}$

$\textcolor{w h i t e}{a a a a}$$\frac{1}{2}$$|$$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$- 15$$\textcolor{w h i t e}{a a a a a a}$$27$$\textcolor{w h i t e}{a a a a a a}$$- 10$

$\textcolor{w h i t e}{a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$- 7$$\textcolor{w h i t e}{a a a a a a a a}$$10$

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

$\textcolor{w h i t e}{a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$- 14$$\textcolor{w h i t e}{a a a a a a}$$20$$\textcolor{w h i t e}{a a a a a a a a}$$\textcolor{red}{0}$

The remainder is $= \left(0\right)$ and the quotient is $= \left(2 {x}^{2} - 14 x + 20\right)$

As the remainder $= 0$, $x = \frac{1}{2}$ is a root of $2 {x}^{3} - 15 {x}^{2} + 27 x - 10$

$2 {x}^{3} - 15 {x}^{2} + 27 x - 10 = \left(2 {x}^{2} - 14 x + 20\right) \left(x - \frac{1}{2}\right)$