How many root passwords should you have?
There's a simple answer to the question of how many root passwords you should have; clearly, you should have a separate root password for each system. This answer is, shall we say, naive in most situations.
We can see why by asking the traditional security question of what the actual risks are of using the same root password on different systems, which is that an attacker who gets your root password from one system can then immediately compromise another. So the first situation where it is mostly or entirely pointless to have separate root passwords is where an attacker could compromise the other machine even without the root password.
The next situation is where blocking the attacker getting root on other machines isn't actually protecting anything meaningful, for example if you use ordinary NFS and the attacker gets root on a machine with enough NFS mount permissions. The attacker hardly needs to get root on any other machine, because they already have full access to user files that are visible from their machine, which in many cases is 'all of them'.
(Sure, NFS doesn't give them access as root, but this is hardly an obstacle; they can use root powers to become the user's UID and then go to town.)
I could go on, but there's a more general principle here: you don't want to think about machines, you want to think about security domains. There is very little point in using different root passwords on machines in the same security domain, and even if you have multiple security domains you may still want to use the same root password across them, because there are some risks to having lots of passwords.
(And you want to think realistically about what is and isn't in each of your security domains. You may conclude that things are intertwined enough that you only really have one security domain, although you could technically argue that you have several.)