I'm studying for my PhD prelim exam in complex analysis, and I ran into this example problem.
Show that the polynomial $$p(z)=z^{47} − z^{23} + 2z^{11} − z^5 + 4z^2 + 1$$ has at least one root inside the unit disk $\{\lvert z\rvert <1\}$.
My initial thought was to try to use Rouché's theorem, but none of the terms dominate all of the other terms on the unit circle, so I can't apply Rouché in an obvious way.
Alternatively, we know that the product of all of the roots must equal 1. So either all of the roots have unit modulus or there is at least one root inside the unit circle. But I can't figure out how to show that at least one root does not have unit modulus.
What am I missing?