If I understood your proposed system correctly, the following seems to be a deal-breaking weakness: Given a number of polynomials representing encrypted message

If I understood your proposed system correctly, the following seems to be a deal-breaking weakness: Given a number of polynomials representing encrypted messages, (x-z) is a common factor of all of them. The GCD of a set of polynomials can very efficiently be computed. Thus, an attacker observing different messages encrypted with the same key over time gets a better and better idea of what the key is (how quickly depends on the messages: two plaintext messages that happen to give coprime polynomials M_1(x) and M_2(x) would be enough to get the key).