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.
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The 'x' you have mentioned here is called 'ω' (cube roots of unity).
ω = (-1 + i )/2
on calculation,
ω^2 = (-1 - 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) )
ω = (-1 + i )/2
on calculation,
ω^2 = (-1 - 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) )
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