The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with (sometimes implicit) multiplicity".
If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".
Multiplicity of a prime factor 
In the prime factorization, for example,
- 60 = 2 × 2 × 3 × 5
the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has 4 prime factors, but only 3 distinct prime factors.
Multiplicity of a root of a polynomial 
Let F be a field and p(x) be a polynomial in one variable and coefficients in F. An element a ∈ F is called a root of multiplicity k of p(x) if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)ks(x). If k = 1, then a is called a simple root.
For instance, the polynomial p(x) = x3 + 2x2 − 7x + 4 has 1 and −4 as roots, and can be written as p(x) = (x + 4)(x − 1)2. This means that 1 is a root of multiplicity 2, and −4 is a 'simple' root (of multiplicity 1). Multiplicity can be thought of as "How many times does the solution appear in the original equation?".
The discriminant of a polynomial is zero if and only if the polynomial has a multiple root.
Behavior of a polynomial function near a root in relation to its multiplicity 
Let f(x) be a polynomial function. Then, if f is graphed on a Cartesian coordinate system, its graph will cross the x-axis at real zeros of odd multiplicity and will bounce off (not go through) the x-axis at real zeros of even multiplicity. In addition, if f(x) has a zero with a multiplicity greater than 1, the graph will be tangent to the x-axis, in other words it will have slope 0 there.
In general, a polynomial with an n-fold root will have a derivative with an (n−1)-fold root at that point.
In complex analysis 
Let z0 be a root of a holomorphic function ƒ , and let n be the least positive integer such that, the nth derivative of ƒ evaluated at z0 differs from zero. Then the power series of ƒ about z0 begins with the nth term, and ƒ is said to have a root of multiplicity (or “order”) n. If n = 1, the root is called a simple root (Krantz 1999, p. 70).
We can also define the multiplicity of the zeroes and poles of a meromorphic function thus: If we have a meromorphic function ƒ = g/h, take the Taylor expansions of g and h about a point z0, and find the first non-zero term in each (denote the order of the terms m and n respectively). if m = n, then the point has non-zero value. If m > n, then the point is a zero of multiplicity m − n. If m < n, then the point has a pole of multiplicity n − m.
See also 
- Zero (complex analysis)
- Set (mathematics)
- Fundamental theorem of algebra
- Fundamental theorem of arithmetic
- Algebraic multiplicity and geometric multiplicity of an eigenvalue
- Frequency (statistics)
- Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.