Peonies are divided by cutting the roots into sections that contain 2-5 eyes. It is typical to actually cut-away a great deal of the deeper roots in order to encourage what remains to start new, adaptive growth. Anyhow, you needn't worry too much about wounding the roots.
The way most peony roots are recovered is to dig around the root just beyond the drip-line of the leaves (or where they had been) and to loosen the soil all the way around. Try to get 8" to 10" down. From there, a garden fork is used to lever all the way around the root system, going around and evenly lifting it from the hole. You will probably hear some of the roots pop and break - just remember that the important stuff is all up in the top 4" of the soil and that deeper roots aren't critical - be gentle but firm.
If you do decide to divide the plants, allow the recovered root to rest for a few hours. It will become easier to manipulate and to cut. Use a clean, sharp blade for making the divisions. After you've divided, soak the divisions in a bucket of water for an hour or two and then plant them.
There are probably a dozen or more videos on youtube about dividing peony roots - several of them show the recovery of the root and you might want to watch those.
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