Multiplicity

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	set theory
	mathematical analysis
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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 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 a called a ''simple root''. For instance, the polynomial $p(x)=x^3+2x^2-7x+4$ 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 (multiplicity $1$). MULTIPLICITY OF A ZERO OF A FUNCTION Let $I$ be an interval of R, let $f$ be a function from $I$ into R or '''C''' be a real (resp. complex) function, and let $c$ ∈ $I$ be a zero of $f$, i.e. a point such that $f(c)=0$. The point $c$ is said a ''zero of multiplicity $k$'' of $f$ if there exist a real number $l$ ≠ $0$ such that