If $x = 2+2^{\frac{1}{3}} + 2^{\frac{2}{3}}$, then find the value of $x^3 - 6x^2 + 6x$.
Answer & explanation
Correct answer: option 4
( a + b ) = a3 + b3 + 3ab( a + b )
x = 2 + 22/3 + 21/3
= (x – 2) = (22/3 + 21/3)
Cubing on both sides
= x3 – 8 – 6x2 + 12x = (22/3)3 + (21/3)3 + 3 × 22/3 × 21/3(22/3 + 21/3)
= x3 – 14 – 6x2 + 12x = 6 (x – 2)
= x3 – 14 – 6x2 + 12x – 6x + 12 = 0
= x3 – 6x2 + 6x = 2