Generally you want to leave polyps in place on both resulting pieces.
For example, you can't just hack down a Colt Coral at the base and expect the stump to grow into a new one... but if you cut a branch off at the main stalk, you should be fine.
Yes, the fragged pieces will eventually grow into a full sized coral.
A bubble coral is an LPS (Large Polyp Stony). It is seriously harmful to the coral to attempt to frag it as it leaves a gaping hole that can get infected. It and other LPS like plates, tounges, fungia, brains, etc are best left in one piece. LPS's that branch, like branching frogspawn, candy cane, etc are all easiy fragged, but that's because you can easily break off entire whole polyps.
Never try to divide a single polyp.
Zoos and mushrooms ... keep in mind, nothing NEEDS to be fragged unless it is overgrowing, etc. Really though these are just bunches of individual corals (polyps in this case) living together in a colony. Fragging them is more or less tearing off one polyp and attaching it somewhere else.
