  We show that if $f\in \mathcal{O}_X(X)$ and $g\in \mathcal{O}_Y(Y)$ are
nonzero regular functions on smooth complex algebraic varieties $X$ and $Y$,
then the Bernstein-Sato polynomial of the product function $fg \in
\mathcal{O}_{X\times Y}(X \times Y)$ is given by $b_{fg}(s)=b_f(s)b_g(s)$. This
answers a question of Budur in \cite{Bud12} and of Popa in \cite{Pop21}.
