Healthy roots should be white not black. What kind of philo do you have? Philos can live in water for a long time before potting it up. I'd wait until the roots got about an inch long and start to feather out before potting in a free flowing mix.

Oh no! It's a heart-leaf philodendron. Its mother plant is very healthy. (Or so I thought.) What do I do about the black roots? Does the mother plant have root rot?

More than likely the mother plant is fine if it looks healthy, sometimes cuttings will not take despite our best efforts, then we can try again in a few weeks and have success. As long as the stems are not too mushy all is not lost. Just cut off the mushy (if any) part and try again. The roots (if mushy) can also be cut off, it will attempt to grow more. If the roots are not mushy, let them grow and see what they do.

Thank you for your advice. The roots aren't mushy, they're pretty strong, and the mother plant is still growing. I'm gonna stick it out and see how the cutting does.

Heart-leaf philodendrons often try to grow aerial roots (in fact I'm not sure they ever don't try to grow aerial roots) at the nodes. If the aerial roots don't find moisture, they eventually dry up and go black. I'm guessing what happened is that the plant did this and then the cuttings were taken after the aerial roots dried up. The plant can still root from that node even after this has happened, I'm pretty sure; you just have to wait for it to grow roots that aren't, you know, dead.

If, on the other hand, the roots started out healthy and then went black once they were put in water, that likely means something different.

Either way, there's no rule that says roots have to be this long before you can plant in soil. Some people would skip the water-rooting part entirely and go straight to soil at the beginning. I personally have some difficulty with that (rot being the difficulty), so I start them in water and then move them to soil whenever I have time, which usually means that by the time I plant them the roots are a good three or four feet long.

(Kidding. Usually two or three inches is plenty.)