Multiplicative Thom-Sebastiani for Bernstein-Sato polynomials

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 $b_{fg}(s)=b_f(s)b_g(s)$, answering a question of Budur and Popa.