Let's start small here:
If we have a 2nd degree polynomial (that's a quadratic, baby!), what are all the possible combinations of types of zeros that can happen?
-- 2 real zeros:
*This includes the kissing kind like
the real zeros are 0 AND 0!
0 has a multiplicity of two!
-- 2 complex zeros:
-- 1 complex and 1 real?
No way! The complex guy needs his conjugate buddy.