Using 3D positions and kinematics of stars relative to the Sagittarius (Sgr) orbital plane and angular momentum, we identify 166 Sgr stream members observed by the Apache Point Observatory Galactic Evolution Experiment (APOGEE) that also have Gaia DR2 astrometry. This sample of 63/103 stars in the Sgr trailing/leading arm are combined with an APOGEE sample of 710 members of the Sgr dwarf spheroidal core (385 of them newly presented here) to establish differences of 0.6 dex in median metallicity and 0.1 dex in [$alpha$/Fe] between our Sgr core and dynamically older stream samples. Mild chemical gradients are found internally along each arm, but these steepen when anchored by core stars. With a model of Sgr tidal disruption providing estimated dynamical ages (i.e., stripping times) for each stream star, we find a mean metallicity gradient of 0.12 +/- 0.03 dex/Gyr for stars stripped from Sgr over time. For the first time, an [$alpha$/Fe] gradient is also measured within the stream, at 0.02 +/- 0.01 dex/Gyr using magnesium abundances and 0.04 +/- 0.01 dex/Gyr using silicon, which imply that the Sgr progenitor had significant radial abundance gradients. We discuss the magnitude of those inferred gradients and their implication for the nature of the Sgr progenitor within the context of the current family of Milky Way satellite galaxies, and suggest that more sophisticated Sgr models are needed to properly interpret the growing chemodynamical detail we have on the Sgr system.
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