Wade through the tangled mess of Facebook notification settings
Facebook likes to do a lot of "opt out" rather than "opt in" changes on its app updates, and the most recent culprit is a new persistent notification for users to keep up with their Facebook notifications at all times. The notification will pop up when you first install the new version of the Facebook app, and will stay in your bar until you tap part of it to enter the app or tap a small "x" on the right side to dismiss it, much like notifications from media players. Unfortunately, it will keep coming back if you don't go change the settings in the Facebook app. To disable it, open your Facebook app, tap "menu" (or the soft menu key) and tap into the Facebook settings. Scroll down to the very bottom of the list, where under "Advanced Notification Settings" you should see an option to uncheck "Ongoing notifications". (Some users report not seeing the setting, which we have yet to figure out the reason behind.)
While you're in there, you can also choose to disable any other notification from Facebook. This will be especially important if you happen to like the ongoing notification, because if you keep the other notifications turned on you will receive two Facebook notifications any time that something happens. We know, Facebook's settings are a mess, but if you're going to keep the app installed the least you can do is keep the notifications in check.