Do I actually need a battery?

I am trying to trouble shoot my MacBook Pro. I think the computer is somewhere between 3-5 years old. For about the last year or so the battery has been getting bad. At first it wouldn't charge all the way to 100%. Then it slowly got worse. And one day I checked it to see that it was saying I needed to replace the battery. This computer is one that I bought for my mother. So I dont use it everyday. Well it was ok as long as it was left plugged into the power adapter. Then one day my mother says that the computer started to shut off by itself and she had to keep restarting it. And again a few days later it just stopped working at all. I know the Battery is totally junk. So the question is...

Can I just take the battery out and leave it plugged into the power adapter and it should work? Or does it still need the battery to work. I ask this because it wont startup and I dont really know if its just the fact that the battery is junk or possibly something else. I dont want to buy a new battery only to find out that it still wont start up and there is something else wrong with it...

Thanks