 Solution

On the unit circle, $z^{-1}=\overline{z}$, soThus the trigonometric polynomial is nonnegative, with a zero at $z=-1$, andEquivalently, writing $z=e^{i\theta }$ gives $p(z)=2+2\cos \theta =4{\cos }^2(\theta /2)$. The factor has polynomial degree one as required. Multiplication of $q$ by any constant of modulus one leaves the factorization unchanged; uniqueness of $q$ is not claimed.

This is the simplest instance of the Fejér–Riesz theorem, where nonnegativity of a trigonometric polynomial on the unit circle admits a polynomial modulus-square factorization.