# Multiplicity (mathematics)

In mathematics, multiplicity is a general term meaning "the number of values for which a given condition holds." 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.

## Multiplicity of a root of a polynomial

A number ${\displaystyle a\in \mathbb {R} }$ 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) = (x - a)ks(x).

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

## Example

The following polynomial p:

p(x) = x3 - x2 - x + 1

has 1 and −1 as roots, and can be written as:

p(x) = (x + 1)(x - 1)2

This means that x = 1 is a root of multiplicity 2, and x = −1 is a 'normal' root (multiplicity 1).

## In complex analysis

Let ${\displaystyle n_{0}}$ be a root of a function f, and let n be the least positive integer m such that

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

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).