multiplicity1

Multiplicity refers to the number of times a particular element, root, or factor appears within a given set or equation. In mathematics, this concept is most frequently encountered in the context of polynomial roots and the factors of integers.

For a polynomial, a root can have a multiplicity greater than one. If a factor (x - r) appears n times in the factorization of a polynomial, then r is a root with multiplicity n. For example, the polynomial x^2 - 2x + 1 can be factored as (x - 1)(x - 1), or (x - 1)^2. Here, the root x = 1 has a multiplicity of 2 because the factor (x - 1) appears twice. When a root has an even multiplicity, the graph of the polynomial touches the x-axis at that root but does not cross it. If the multiplicity is odd, the graph crosses the x-axis at the root.

In number theory, multiplicity is used when discussing the prime factorization of an integer. If a prime number p appears k times in the prime factorization of an integer n, then p has a multiplicity of k in the factorization of n. For instance, the prime factorization of 12 is 2 x 2 x 3, which can be written as 2^2 x 3^1. The prime factor 2 has a multiplicity of 2, and the prime factor 3 has a multiplicity of 1.