Christmas Cacti are notoriously difficult to root. This article might help you in rooting pieces for new plants or just rooting another in the same pot with the parent plant.
Sometimes a large Christmas cactus will start to fall apart. In this case you can take the large parts that have fallen apart from the big plant and root them. You can also try cutting a large part off of a big cactus and rooting it. In any case, you need a big piece of cactus to root a new plant. The picture of the cactus is one that I rooted. It is in the pot with the parent plant. The piece I rooted is about 10" wide and 7" tall. Smaller piece usually don't root.
Place the piece of cactus in a really good potting soil for indoor plants. Use new, store bought, potting soil. Place the piece into the pot with the soil and lightly pack the soil around the stem until the piece will stand up on it's own. Always take a cutting that has at least three leaves for the stem. Bury at least one leaf under the soil. The stem needs to be long enough to grow roots from.
Keep the soil moist, but not wet. Christmas cactus are succulents and not a true cacti. They grown on the floor of forests and in tree hollows and arms where it is moist. So don't let the soil dry out completely.
The cutting will wilt and shrivel and look like it is going to die. This always happens. The cutting might die, but just keep caring for it.
When the cutting finally starts to take root, it may have been two months or so. You will notice that the stalk or stem has greened up. The outer leaves will still be shriveled and wilted. Just keep watering the plant. Just leave the wilted leaves alone even though you may want to cut them off. The leaves are alive and as the root system grows they will green up and puff up into healthy leaves.