2014/0042) show that x = 2 + 2^(1/3) + 2^(2/3) is a root of x^3 - 6x^2 + 6x -2 = 0

Let x = 2 + 2^(1/3) + 2^(2/3)
so x -  2 =  2^(1/3) + 2^(2/3)  ... (1)

cube both sides to get (x-2)^3 = 2 + 4 + 3 *2 * (2^(1/3) + 2^(2/3)) = 6 + 6 (x-2) ( from 1)
or x^3 - 6x^2 + 12 x - 8 = 6x - 6
or x^3 - 6x^2 + 6x - 2 = 0

so x is a root of above equation
proved