How would one solve $z^6+z^4+z^3+z^2+1=0$?

Question

I recently came upon the following equation, which is supposed to be solvable using complex numbers: $$z^6+z^4+z^3+z^2+1=0$$ The problem is that it is a degree $6$ polynomial with many other exponential terms, which I don't know how to deal with. I try substituting $w=z^2$ to try and make the equation simplier:$$w^3+w^2+wz+w+1=0$$but I can't make any progress with this either. Only thing that I see that could be used is that $w^3,w^2,w,1$ is a geometric series, not sure how that helps though.

Answer 1

If $\omega\neq1$ is a fifth root of unity then $$\omega^6+\omega^4+\omega^3+\omega^2+1=\omega^4+\omega^3+\omega^2+\omega+1=\frac{\omega^5-1}{\omega-1}=0.$$ So we can factor out each of the fifth roots of unity (that aren't $1$) to get $$z^6+z^4+z^3+z^2+1=(z^2-z+1)(z^4+z^3+z^2+z+1).$$ And from here it's easy to see the solutions.

Answer 2

Hint:

Here is general idea how to solve symmetric polynomial equation of degree $6$: Divide it with $z^3$ and you get $$ (z^3+{1\over z^3})+ (z+{1\over z}) +1=0$$ Now write $t= z+{1\over z}$, then $$z^3+{1\over z^3}=t^3 -3t$$ so now you have $$t^3-2t+1=0$$ which has root $t=1$...

Answer 3

Note that neither $z = 1$ nor $z = -1$ are roots. Then our equation becomes equivalent to $$0 = (z^2 - 1)(z^6 + z^4 + z^3 + z^2 + 1) \text{ and } z \neq \pm 1.$$ This equation simplifies to $$z^8 + z^5 - z^3 - 1 = 0.$$ We can factor this by: $$z^8 + z^5 - z^3 - 1 = z^5(z^3 + 1) - 1(z^3 + 1) = (z^5 - 1)(x^3 + 1) = 0.$$ From this, and the fact that $z \neq \pm 1$, you can get the result easily.

How did I get this answer? It was mostly inspired by jlammy's answer, but one of the things that jumped out at me was that $z^3$ was an odd term out of a geometric series $z^6 + z^4 + z^2 + 1$. This meant that multiplying by $z^2 - 1$ would achieve some kind of simplification. I thought it was worth a shot.

In general, if you have a polynomial, whose coefficients are $\pm 1$, and you suspect that some roots of unity are involved, it's probably not a bad idea to try multiplying by $z^2 - 1$ anyway. It should take any of the cylotomic factors like $z^4 + z^3 + z^2 + z + 1$ or $z^2 - z + 1$, and turn them into simpler binomial factors like $z^5 - 1$ or $z^3 + 1$.