Information AboutMultiplicity |
| CATEGORIES ABOUT MULTIPLICITY | |
| 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 correctly, without specifying exceptions (for example, ''double roots'' counted twice). Hence the expression ''counted with (sometimes implicit) multiplicity''. When mathematicians wish to ignore multiplicity they will refer to the number of distinct elements of a set. MULTIPLICITY OF A PRIME FACTOR In the Prime Factorization : 60 = 2 × 2 × 3 × 5 the multiplicity of the prime factor 2 is 2, while the multiplicity of the prime factors 3 and 5 is 1. Thus, 60 has 4 prime factors, but only 3 distinct prime factors. 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'')''k''''s''(''x''). If ''k'' = 1, then ''a'' is called a ''simple root''. For instance, the polynomial ''p''(''x'') = ''x''3 + 2''x''2 − 7''x'' + 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 (of multiplicity 1). The Discriminant of a polynomial is zero if and only if the polynomial has a multiple root. Geometric behavior Let ''f''(''x'') be a Polynomial function. Then, if ''f'' is graphed on a Cartesian Coordinate System , its graph will cross the ''x''-axis at real zeros of odd multiplicity and will 'bounce' from the ''x''-axis at real zeros of even multiplicity. In addition, if ''f''(''x'') has a zero with a multiplicity greater than 1, the graph will appear flatter at values close to it. MULTIPLICITY OF A ZERO OF A FUNCTION Let be an interval of R, let be a function from into R or '''C''' be a real (resp. complex) function, and let ∈ be a zero of , i.e. a point such that . The point is said a ''zero of multiplicity '' of if there exist a real number such that |
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