$z^6+iz^3+i-1=0$

Question

I am trying to solve $z^6+iz^3+i-1=0$. I tried to factor/reduce the equation, but nothing seems to be helpful.

I don't necessarily need the solution to this. But, I want to know what is the right approach to reducing such equations? Where can I practice such problems?

Answer 1

Different approaches:

Look for solutions.

You might be able to spot $z=-1$ as a solution right away. Now we can make progress by factoring out (z+1).

Apply the formulas you know.

I would think that $a^3+b^3=(a+b)(a^2-ab+b^2)$ and $a^2-b^2=(a+b)(a-b)$ would be particularly useful.

Use a tool

You can find the answer here

Your next question asks for practice problems. I would think that AOPS books on algebra and precalculus would do the trick. But any good textbook on Algebra 2/ precalculus content should have practice like this in it.

Answer 2

$$z^6-1+i(z^3+1)= (z^3+1)(z^3-1)+i(z^3+1)$$ $$=(z^3+1)(z^3-1+i)$$

Answer 3

Hint : Letting $x = z^3$, your problem it convert in a quadratic equation $$x^2 + ix + (-1+i) = 0$$ and the left hand side it can be factored as $$(x-(1-i))(x+1)$$