What is the best way to stop my pear tree roots from invading and sucking the life out of my flower bed that is planted under it?
Usually, I don't try to keep trees out of landscaped beds. But if you are going to garden under a tree (pear trees aren't the worst), you'll want to plant a shade garden. Probably a dry shade garden. There are a lot of good ornamentals that will tolerate growing under a tree. I'm not sure where you're located, that would help me recommend something specific. Usually it's also safe to go with a dry woodland plant that's native to the area.
Now, if the tree has not yet grown roots into the area, you can block many of the roots by installing a root control landscape fabric barrier. This usually goes about 12" into the soil, so doesn't control below that level. I rarely use the stuff, as you can very easily work with nature when dealing with trees over beds. You can also use something more substantial, as described by this answer on horsetail control. Again, I don't like intervening where I don't have to.
Trees don't suck the life from flower beds. They use water and nutrients, but both these thing are readded to the soil for orchard/ornamental trees anyway. If you aren't watering, or adding compost or fertilizer, that's another thing you could do.