Question

If x= -1/2+ root3/2 i where i=root of -1. Find the value of the expression:
P (x)= x^4 + 2x^3 + 3x^2 + 2x + 2.

Answer 1

The 'x' you have mentioned here is called 'ω' (cube roots of unity).

ω = (-1 + $\sqrt{3}$ i )/2
on calculation,
ω^2 = (-1 - $\sqrt{3}$ i )/2
and 
ω^3 = 1

so naturally, 

ω^4 = ω  (= ω^3 * ω)              ----------------------(i)         

also, ω + ω^2 + 1 =0              ----------------------(ii)

On transforming the given equation in terms of ω , ω^2 and 1 (ie ω^3),
we get

ω^4 + 2ω^3 + 3ω^2 + 2ω + 2.
= ω + 2 ( 1 ) + 2*(ω^2) + 2* ω +2               ( By (i) )
=2*(ω^2) +2*(ω) +2 
= 2(ω + ω^2 + 1)
=0                                                              ( By (ii) )