Integration-by-parts reductions play a central role in perturbative QFT calculations. They allow the set of Feynman integrals contributing to a given observable to be reduced to a small set of basis integrals and moreover facilitate the computation of those basis integrals. In this talk I introduce an efficient new method for generating integration-by-parts reductions. This method simplifies the task by making use of unitarity cuts and turns the problem of finding the needed total derivatives into one of solving certain polynomial (so-called syzygy) equations.