The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.
How do I intuitively understand multiplicity in the above statement? For example the equation $y=x^2$ is said to have a zero at $0$ and this has multiplicity $2$. In general when can I say that a certain root has a multiplicity $2$ while a different root has multiplicity $4$ vs a root with multiplicity $1$ without factoring?
Are there any prominent generalizations that are used?