[villagetinker]

I agree with removing the offending tree root and installing root barrier. As for the repair, is the driveway finished or unfinished concrete? My point, if unfinished, there are multiple products at the hardware stores that should work for you, however I would make an effort to have the offending root removed if possible and then let the driveway settle. If this is not possible any repair (crack sealer) will probably be a short-term fix. Cutting around the root and removing the driveway is probably the better option but will definitely be the most costly.
If the driveway is finished then i would contact the company that did the finish and ask them if a repair is practical, and then what they would recommend.