# Does $z^4+2z^2+z=0$ have complex roots?

Does $$z^4+2z^2+z=0$$ have complex roots? How to find them? Besides $$z=0$$, I got the equation $$re^{3i\theta}+2re^{i\theta}=e^{i(\pi+2k\pi)}$$, $$k\in \mathbb Z$$. How to find the complex roots?

• By the fundamental theorem of algebra, every polynomial of degree $n$ has exactly $n$ roots in $\mathbb{C}$. As for finding yours, though, I'm not sure. – Eevee Trainer Nov 29 '18 at 3:01
• The polynomial is $zP$ where $P = z^3+2z+1$. Since $\frac{dP}{dz} = 3z^2 + 2$ is positive for all $z$, it follows that $P$ is an increasing function and can have only one real root. – Théophile Nov 29 '18 at 3:17