Whether or not a fractured tooth can be salvaged depends on the orientation and extent of the fracture, and the overall health of the tooth, supporting tissues, and the context in which it is situated.
Strictly speaking, any tooth can be "saved"-- the more important question is whether the time, effort, and expense of repair will be justified by the quality of the result.
For example, if a crack runs deeply under the gum, the more superficial piece of tooth above the crack can be removed, the gum cut away to expose the root remnant in the gum, and a crown
can be constructed to extend that root fragment up to where it can contact the opposing tooth during chewing. However, if too much of the tooth is damaged by the fracture, the amount of root that remains embedded in the gum and bone will be insufficient to sustain the considerable forces generated by chewing. Such a tooth would be loose, uncomfortable to chew on, and either unconsciously or consciously, you would be making a concerted effort to avoid chewing on it. So, in an effort to save the tooth, you would actually be diminishing the functionality of that part of your mouth.
In another example, a crack in a molar with two roots can effectively separate the roots from each other, creating two independent roots. Each of these roots can have a separate bicuspid-type crown fabricated for it, creating two teeth where there was one. However, the relative positions of the roots would need to be such that the crown contours wouldn't create a periodontal
problem, as is often the case when this "bicuspidation" procedure is attempted. When used appropriately, this technique can sometimes work, but it doesn't often have the probability of success that would justify the time or effort.
In sum, it is often advisable to save a tooth if possible, but some teeth are just so badly damaged that a more favorable outcome can be derived by extraction followed by some type of prosthetic replacement- a fixed bridge or implant prosthesis, for example. Sadly, these fractured-tooth situations are often the type of settings where discretion must be the better part of valor. In other words, a cracked root, more often than not, will severely compromise a tooth's ability to function reliably in the future, and it is often best to know when extraction is not such a bad idea.
Hope this helps...