It's a general problem with the Facebook app for Android. You'll get a notification from time to time, but usually you have to launch the app on a regular basis to check it. They keep trying to patch it, and sometimes it works for a bit, but usually stops. I've just grown used to it over time and I'm sure you will too.
Have a friend with a blackberry? log into your fb account using their phone in their fb app... lave it be for about 2 minuted and just post something on your wall from another persons account (anything that would push a notification) and tada! you now have notifications on your android fb now you can log out of your account from your buddy's bb and wash your hands