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.