How to determine a system is stable using pole zero analysis?
For a LTI system to be stable, it is sufficient that its transfer function has no poles on the right semi-plane.

Take this example, for instance: F = (s-1)/(s+1)(s+2). It has a zero at s=1, on the right half-plane. Its step response is:


As you can see, it is perfectly stable.

The characteristic function of a closed-looped system, on the other hand, cannot have zeros on the right half-plane. The characteristic function of a closed loop system is the denominator of the overall transfer function, and therefore its zeros are the poles of the system. That's why you are mixing things up.

A very important concept, worth mentioning, is closely related with the existence of zeros on the right half-plane, though: minimum and maximum phase systems. I suggest you take a look at the wikipedia article about it.