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First principles electronic structure and transport calculations are used to demonstrate the impact of the electric polarization on electron and spin transport in Fe/BaTiO3/Fe multiferroic tunnel junctions (MFTJs). We find that the polarization of BaTiO3 reduces the tunneling conductance, as compared to a nonpolarized barrier, due to the change in the electronic structure driven by ferroelectric displacements, similar to that found previously for Pt/BaTiO3/Pt. For the MFTJ, however, this effect has different magnitudes for majority- and minority-spin channels and for parallel and antiparallel orientations of the magnetization of the electrodes. As a result, we find a substantial drop in the spin polarization of the tunneling current in the parallel configuration and an inversion of the magnetoresistance as polarization of the barrier is turned on.