  This paper contains sharp bounds on the coefficients of the polynomials $R$
and $S$ which solve the classical one variable B\'{e}zout identity $A R + B S =
1$, where $A$ and $B$ are polynomials with no common zeros. The bounds are
expressed in terms of the separation of the zeros of $A$ and $B$. Our proof
involves contour integral representations of these coefficients. We also obtain
an estimate on the norm of the inverse of the Sylvester matrix.
