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.