# How do you use synthetic division to find all the rational zeroes of the function f(x)= 2x^3 - 3x^2-11x+6?

Aug 20, 2015

$3 , - 2 , \frac{1}{2}$

#### Explanation:

A rational zero must be ± frac{\text{a divisor of 6}}{\text{a divisor of 2}} = ± frac{{1,2,3,6}}{{1,2}} \in ± {1,2,3,6, 1/2, 2/2, 3/2, 6/2}

$f \left(1\right) = 2 - 3 - 11 + 6 < 0$

$f \left(2\right) = 16 - 12 - 22 + 6 < 0$

$f \left(3\right) = 54 - 27 - 33 + 6 = 0$

By Briot-Ruffini,

$\setminus \frac{f \left(x\right)}{x - 3} = 2 {x}^{2} + 3 x - 2 = 0$

$\setminus \Delta = 9 + 4 \cdot 2 \cdot 2 = 25$

x = frac{- 3 ± 5}{4}