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Magnetically tunable Feshbach resonances in ultracold atomic systems are chiefly identified and characterized through time-consuming atom loss spectroscopy. We describe an off-resonant dispersive optical probing technique to rapidly locate Feshbach resonances and demonstrate the method by locating four resonances of 87Rb, between the |F=1,mF=1⟩ and |F=2,mF=0⟩ states. Despite the loss features being ≲0.1 G wide, we require only 21 experimental runs to explore a magnetic field range >18 G, where 1G=10-4T. The resonances consist of two known s-wave features in the vicinity of 9 G and 18 G and two previously unreported p-wave features near 5 G and 10 G. We further utilize the dispersive approach to directly characterize the two-body loss dynamics for each Feshbach resonance. [ABSTRACT FROM AUTHOR]