Generally speaking we are told that plants that have a tap root cannot be divided. A taproot is a long slender fleshy root that goes straight and deeply into the earth. A carrot is a modified taproot. It is designated modified because it is also a food storage area and is generally thicker than the average taproot.
The most skilled gardener isn’t going to try to divide a carrot to make two plants however think what happens when a carrot is grown in heavy soil or the taproot hits a rock. Under these conditions a taproot will divide. The configuration that results is more common in other taproot plants such as the butterfly weed than it is in domestically grown carrots. The older the plant the more likely it is to develop these side root systems. A taproot may be divided at these divisions if they also contain some fibrous roots.
The divisions near the surface may have some foliage attached. They can be cut using a sharp clean knife with the foliage attached and another ready-made plant results. Further down on the root the divisions are less likely to have foliage but they may have an eye. This is an area of the plant that, while dormant, is capable of producing foliage. Cutting of the side root complete with fibrous roots and an eye will in time produce another plant.
Younger plants may have a single taproot but it is rare for this condition to continue as the plant ages. This tendency does make it possible to divide a plant with a taproot. The catch in this procedure is that the plant must be totally removed from the ground to perform the division. This means that the entire taproot must be dug up and these can go quite deeply into the soil. Breaking off a taproot may not be fatal to a plant but it probably doesn’t do it any good. It also provides an area where bacteria or fungi can enter the plant and cause decay. The result may be the loss of the parent plant. Dividing a plant with a taproot is possible but not an exercise for the timid.