If `x=omega-omega^2-2` then , the value of `x^4+3x^3+2x^2-11x-6` is (where `omega ` is a imaginary cube root of unity)

Answer

1

Explanation

We have `x= omega - omega^(2) - 2 or x + 2 = omega -omega^(2)`
Squaring, `x^(2) + 4x + 4 = omega^(2) + omega^(4) - 2omega^(3) = omega^(2) + omega -2=-3`
`rArr x^(2) + 4x + 7 =0`
Dividing `x^(4) + 3x^(3) + 2x^(2) -11x - 6 by x^(2) + 4x +7`, we get
`x^(4) + 3x^(3) + 2x^(2) - 11x -6 = (x^(2) + 4x + 7) (x^(2) -x-1)+1`
`= (0) (x^(2) -x-1)+ 1=0 +1 =1`