  We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on
the Hardy space, in terms of the distance to its spectrum $\sigma(T_b)$. We are
primarily interested in the case when the symbol $b$ is a Laurent polynomial
(\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such
symbol the growth of the resolvent is quadratic, and under certain additional
assumption it is linear. We also prove the quadratic growth of the resolvent
for a certain class of non-rational symbols.
