[Math] Show that $z^6 + 5z^4 – z^3 + 3z$ has at least two real roots given that all roots are distinct.

Question

Show that $z^6 + 5z^4 – z^3 + 3z$ has at least two real roots given
that all roots are distinct. Also, show that $|3z – z^3 + 5z^4| < |z^6|$ when $|z| > 3$.

I can see that 0 is a real root; however, I am having trouble starting this one. I couldn't seem to see a way to factorize this. Other than testing points in the derivative and maybe using the intermediate value theorem, I don't know what direction to take on this one.

Answer 1

Since the coefficients are real, all roots have conjugate roots as well. This means that excluding the $z=0$ root, you have five other distinct roots, so at least one of them is real. So we have at least two real roots.