TITLE: Methods for determining which roots of a polynomial are inside of the unit circle?
QUESTION [6 upvotes]: 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.

REPLY [7 votes]: As rogerl said, by symmetry the number of roots inside the unit circle is equal to the number outside.  Now, how many are on the unit circle?  Let's suppose $1$ is not a root, i.e. $2 + 2 b + c \ne 0$.  The Möbius transformation
$ w = i(1+z)/(1-z)$ ($z = (w-i)/(w+i)$) takes the unit circle (except for the point $1$) to the real line, and $p(z) = 0$ becomes $g(w) = (2b+c+2) w^4+(2c-12)w^2-2b+c+2 = 0$.
The real solutions of $g(w) = 0$ correspond (in pairs, when nonzero) to nonnegative solutions of
$(2b+c+2) t^2 + (2c-12) t - 2b + c + 2 = 0$.  The roots of that quadratic are 
${\dfrac {-c+6\pm 2\,\sqrt {{b}^{2}-4\,c+8}}{2\,b+c+2}}$.  So for example when 
$b = 1$ and $c = 2$, the quadratic has two positive roots ($1$ and $1/3$), and all four of the roots of your quartic are on the unit circle.