Is there a way to change passwords on multiple servers simultaneously?

You could also use PsPasswd from the SysInternals suite. PsTools tend to have a few dependencies (like maybe a default share or something) but they're usually pretty stable. It's worth noting it does send passwords in the clear. I don't know if the other solution does too.

If the servers are on the domain.

Local Administrator Password Solution

Microsoft is offering the Local Administrator Password Solution (LAPS) that provides a solution to the issue of using a common local account with an identical password on every computer in a domain. LAPS resolves this issue by setting a different, random password for the common local administrator account on every computer in the domain. Domain administrators using the solution can determine which users, such as helpdesk administrators, are authorized to read passwords.

https://technet.microsoft.com/en-us/library/security/3062591.aspx