Question

Valoarea lui (1 − α)(1 − α^2)(1 − α^4)(1 − α^5), unde α ∈ C \ R, α^3 = 1, este?

Answer 1

α³=1

α3-1=0

(α-1) (α²+α+1)=0

deci

α1=1 ∈R

α²+α+1=0

α2,3=(-1/2+-√3)/2

cu α2=α1 conjugat=(α1)²

cum α1∈R, innseam na ca e vorba de α2 sauα3

tinand cont ca (α1)²=α2 si (α2)²=α1, iar α³=1, expresia devine

(1-α)(1-αconjugat)(1-α)(1-αconjugat)=((1-α1)(1-α2))²= (1-(α1+α2)+α1α2)²= (viete)=

(1-(-1)+1)²=3²=9