how do we do something like this question
z^6 + 8z^3 - 3=O
There is no universal solution for the sixth degree polynomial equation. However, for this particular equation you can substitute a=z^3
so you will end up with quadratic equation a^2+8a-3=0
Solve equation to find roots -4+/-sqrt(19), or
a = 0.3589
a = -8.359.
Then you will have to take third root from both of these numbers to get two answers
-2.0295
0.7107.
Edit: I noticed that the question was posted in the “complex numbers”, that means we have to calculate all six roots of the equation, including imaginary roots (the amount of roots of a polynomial equation is always equal to the degree of the equation, however, some roots can be imaginary).
To find all roots we will have to apply De Moivre’s theorem:
