  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.
