Prove that the roots of a given polynomial don't have the same absolute value

Question

Let $n \geq 4$ and $f=\sum_{k=0}^{n}a_{k}X^{k}$ be a polynomial with complex coefficients, such that $a_{n}=1,$ $a_{1}\cdot a_{n-1}\neq 0$ and $|Re(a_{n-1})|>\sqrt[n-2]{\left | \frac{a_{1}}{a_{n-1}} \right |}.$
Prove that $f$ does not have all of its roots of the same absolute value.

I haven't done anything meaningful yet. Thank you in advance!

Answer 1

This is false.

Let $n=4$ and consider $f(x)=x(x-1)^2(x-2)=x^4-4x^3+5x^2-2x$. All hypotheses hold:

• $a_n=1$
• $a_{n-1}=-4$ and $a_1=-2$, so their product is nonzero
• $|\Re(a_{n-1})|=4 > \sqrt{\left|\frac{a_1}{a_{n-1}}\right|}=\frac{\sqrt2}2$

By construction, $f$ has a double root $1$, so it is a counter-example.