[Math] Methods for determining which roots of a polynomial are inside of the unit circle

Question

Let's say I have a polynomial such as $$p(x) = x^4 + bx^3 + cx^2 + bx + 1.$$

I strongly suspect that, for any parameters, there are always two roots inside the unit circle and two roots outside of the unit circle.

What tools can I use to determine whether or not this is correct?

I am not necessarily looking for a solution to this problem but any answer that solves this problem would naturally contain tools that can be used in the general case.

Thank you.

Answer 1

$p\left(\frac{1}{x}\right) = \frac{1}{x^4}p(x)$. So unless there are roots on the unit circle (which is not ruled out in the problem as stated), there are two inside and two outside the unit circle.