Some cultivated herbaceous peonies can form adventitious buds on roots well away from the crown tissue. Adventitious buds develop where you wouldn't normally expect them; a common example is roots growing from stem cuttings. This means that if you are dividing a peony cultivar known to have adventitious buds, you don’t need root pieces with buds on them to generate new plants. Save the other root pieces and plant them too. Note that it may take a few more years for this root to become a mature flowering plant.
A list of cultivars forming adventitious buds (PDF) is available on the American Peony Society website.