  Let $f$ and $F$ be two polynomials satisfying $F(x)=u(x)f(x)+v(x)f'(x)$. We
characterize the relation between the location and multiplicity of the real
zeros of $f$ and $F$, which generalizes and unifies many known results,
including the results of Brenti and Br\"and\'en about the $q$-Eulerian
polynomials.
