Originally Posted by
Olym Hi everyone, this is my first post here. If this belongs in a different section, my apologies.
Question: P(z) := z^3 + pz + q, p and q are elements of Real numbers, u and v and Ϛ are elements of the complex numbers.
u^3 = (-q/2) + sqrt((q/2)^2 + (p/3)^3) and v^3 = (-q/2) - sqrt((q/2)^2 + (p/3)^3)
and Ϛ^3 = 1
Show that z1 = u + v, z2 = Ϛu + (Ϛ^2)v and z3 = (Ϛ^2)u + Ϛv fufill the condition that P(z1) = P(z2) = P(z3) = 0
I have the following:
u^3 + v^3 = -q
and (u*v)^3 = (u^3)*(v^3)
= ((-q/2) + sqrt((q/2)^2 + (p/3)^3))*((-q/2) - sqrt((q/2)^2 + (p/3)^3))
= ((-q^2)/4)) - ((p^3)/27)
and
P(z1) = (u+v)^3 + p(u+v) + q
= u^3 + v^3 + 3((u^2)*v + u*(v^2)) + p(u+v) + q
= 3((u^2)*v + u*(v^2)) + p(u+v)
I don't know how to continue.
Thanks for any help.