Let be a polynomial of degree with real coefficients for
Show that if for some then or .
I'm kind of stuck on this one.
I observed that there are clearly no roots on the positive real axis. I could use Descarte's Rule of Signs if necessary.
I also know, via the same method described here, that all roots have modulus less than or equal to 2.
I was thinking of writing and using the statement that to get some relationship between and using trig functions but it hasn't panned out for me so far.
I also wonder if Vieta's formula for the coefficients in terms of the roots might help.