Question

What is the relationship between the degree of a polynomial and the number of zeros (roots) it posses? Be sure to explain in your answer how prime polynomials violate this relationship.

Answer 1

If I remember correctly, a n-th degree polynomial has exactly n roots, although they may be real or complex. A n-th degree polynomial will have at most n real roots.
A prime polynomial of degree n will not have n real roots, for example:
f(x) = x^2 + 4 has no real roots (try to factor it), but 2 complex ones, namely +/- 2i