I've read that it's quite common among students to take a graduate program which is different from the bachelor's degree. I was wondering if it is possible for someone with a bachelor's degree in Political Science and Master of Business Administration to teach Political Science subjects.
Highly unlikely. Even at the community college level, to teach in a field one must typically have at least some postgraduate degree in that field or in a closely related field. So you someone probably teach in a polisci department with a degree in economics, because they use many of the same methods and consider some of the same areas. Ditto philosophy. Not a professional business degree.
(PhD political scientist here)
It is possible for people to teach across disciplines if their record reflects knowledge in the course they will teach. With a BS and PhD in Aerospace Engineering I have taught courses in CS and a division of Statistics and Scientific Computation. I was eligible to do this because my background is in parallel computing for computational fluid dynamics problems, so I have demonstrated skills in programming, software engineering, numerical analysis, and parallel computing. The courses I was teaching were on scientific computing and parallel computing and were being offered first by a CS department and then by this division (which has now become a department). I was deemed qualified to teach them based on my background and demonstrated skills even though my PhD is in a different subject.
So yes, it's possible, but I think you'd have to show some background related to the subject in question. It will also depend on what kind of job you are applying for. If a Poli Sci department needs an adjunct for one course, you might qualify to teach that course, but you are unlikely to land a full-time position unless your Doctor of Business Administration research was related to politics somehow.