  This paper deals with sequences of random variables $X_n$ only taking values
in $\{0,\ldots,n\}$. The probability generating functions of such random
variables are polynomials of degree $n$. Under the assumption that the roots of
these polynomials are either all real or all lie on the unit circle in the
complex plane, a quantitative normal approximation bound for $X_n$ is
established in a unified way. In the real rooted case the result is classical
and only involves the variances of $X_n$, while in the cyclotomic case the
fourth cumulants or moments of $X_n$ appear in addition. The proofs are
elementary and based on the Stein-Tikhomirov method.
