# 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 is called a *root of multiplicity k* of a polynomial *p* if there exists a polynomial *s* with:

and

*p(x) = (x - a)*.^{k}s(x)

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

## Example

The following polynomial *p*:

*p*(*x*) = x^{3}- x^{2}- 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).

See also Fundamental Theorem of Algebra, separable polynomial, intersection number.

## In complex analysis

Let be a root of a function *f*, and let *n* be the least positive integer *m* such that

- .

Then the power series of about begins with the th term, and is said to have a root of multiplicity (or "order") . If , the root is called a simple root (Krantz 1999, p. 70).