Thanks FlowerName and welcome to The Buzz.
A couple of points upfront: Although they may be a nuisance, the carpenter ants are not seriously or not at all damaging to the tree. They usually exploit wood that has begun to decay or breakdown through the action of wood decay fungi. Their pattern of infestation within the tree follows the pattern of tree compartmentalization of decay. Although there are exceptions, most wood decay infections spread rather slowly in living trees and are introduced through mechanical wounding, especially of the roots and stem. So the ants exploit that niche, they don't create it.
Don't spend much time on trying to seal the entrances/exits. You will not get all of them or even most of them...particularly as some are likely to be belowground.
So what to do? Keeping the tree healthy with the proper amount of water (neither too much nor too little), proper soil drainage, avoidance of soil compaction, avoidance of additional injuries are all good steps.
I tend to avoid chemical insecticides, but if you are going to use them, follow the label directions carefully. Chemical availability has greatly changed in recent years and varies state-by-state. Most insecticides for ant control are tested around human infrastructure and not on living trees, so there may be some phytotoxicity as well. Also learn a little about carpenter ant behavior. They usually have a central colony as well as some satellite colonies, so total eradication can be tough. The key thing here is that eradication of the ant is sort of missing the target!
