what is multiplicity

In mathematics, multiplicity refers to the number of times a member appears in a multiset. In prime factorization, the multiplicity of a prime factor is its p-adic valuation. In the context of polynomials, multiplicity refers to the number of times a zero appears in the factored form of the polynomial. For example, if a polynomial has a zero at x=3 that is related to the factor (x-3)², then x=3 is a zero of multiplicity 2, and x=1 is a zero of multiplicity 1. The concept of multiplicity is directly related to the graphical behavior of the polynomial around the zero. If a zero has a higher than one multiplicity, then there will be fewer distinct zeros. In general, multiplicity refers to the number of times a given condition holds.