Solve $z^6+7z^3-8=0$

Question

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation.

This is an exercise that involves complex numbers, so I have to transform the equation into trigonometric form and then use the de Moivre formula.

Any hints?

Answer 1

$$(z^3 + 8)(z^3 - 1) {}{}{}{}$$

Answer 2

Using Mathematica

poly = z^6 + 7 z^3 - 8;

Factor the polynomial and solve the equations formed by setting each factor equal to zero

factoredPoly = Factor[poly]

(*  (-1 + z)*(2 + z)*(4 - 2*z + z^2)*(1 + z + z^2)  *)

This reduces the problem to solving quadratic and linear equations

soln = Flatten[Solve[# == 0, z] & /@ List @@ factoredPoly, 1]

(*  {{z -> 1}, {z -> -2}, {z -> 1 - I*Sqrt[3]}, 
   {z -> 1 + I*Sqrt[3]}, {z -> -(-1)^(1/3)}, 
   {z -> (-1)^(2/3)}}  *)

Verifying that this is equivalent to solving the original equation

(soln // Sort) == Solve[poly == 0, z]

(*  True  *)