# Multiplicity

In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. For example, the term is used to refer to the value of the totient valence function, or the number of times a given polynomial equation has a root at a given point.

The common reason to consider notions of multiplicity is to count right, without specifying exceptions (for example, double roots counted twice). Hence the expression counted with (sometimes implicit) multiplicity.

## Multiplicity of a prime factor

In the prime factorization

60 = 2 × 2 × 3 × 5

the multiplicity of the prime factor 2 is 2; the multiplicity of the prime factor 3 is 1; and the multiplicity of the prime factor 5 is 1.

## Multiplicity of a root of a polynomial

A real or complex number a is called a root of multiplicity k of a polynomial p if there exists a polynomial s with:

${\displaystyle s(a)\neq 0}$

and

p(x) = (xa)ks(x).

If k = 1, then a is a simple root.

### Example

The following polynomial p:

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 x = 1 is a root of multiplicity 2, and x = −4 is a 'simple' root (multiplicity 1).

## In complex analysis

Let ${\displaystyle z_{0}}$ be a root of a holomorphic function f, and let n be the least positive integer m such that, the m-th derivative of f evaluated in ${\displaystyle z=z_{0}}$ differs from zero:

${\displaystyle f^{(m)}(z_{0})\neq 0.}$

Then the power series of ${\displaystyle f}$ about ${\displaystyle z_{0}}$ begins with the ${\displaystyle n}$th term, and ${\displaystyle f}$ is said to have a root of multiplicity (or "order") ${\displaystyle n}$. If ${\displaystyle n=1}$, the root is called a simple root (Krantz 1999, p. 70).