Complex polynomial and the unit circle

Let $g(z):=z^nP\left(\frac 1z\right)$. It's a polynomial whose leading term is $a_0$ and constant coefficient is $1$. We have that $g(0)=1$ and $\max_{|z|=1}|g(z)|=1$, hence by maximum modulus principle, $g$ is constant equal to $1$. This gives the wanted result.