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In prime factorization, the multiplicity of a prime factor is its -adic valuation.For example, the prime factorization of the integer 60 is . 60 = 2 × 2 × 3 × 5, the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1.
A polynomial equation, ... If P is a nonzero polynomial, there is a highest power m such that (x − a) m divides P, which is called the multiplicity of a as a root of P.
This provides a recursive formula for the cyclotomic polynomial ... As m is the multiplicity of p in n, p cannot divide the value at 1 of the other factors of (). ...
As R is a homogeneous polynomial in two indeterminates, the fundamental theorem of algebra implies that R is a product of pq linear polynomials. If one defines the multiplicity of a common zero of P and Q as the number of occurrences of the corresponding factor in the product, Bézout's theorem is thus proved.
Whereas equation factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity, = () () ().
Kahan discovered that polynomials with a particular set of multiplicities form what he called a pejorative manifold and proved that a multiple root is Lipschitz continuous if the perturbation maintains its multiplicity. This geometric property of multiple roots is crucial in numerical computation of multiple roots.
The roots of the corresponding scalar polynomial equation, λ 2 = λ, are 0 and 1. Thus any projection has 0 and 1 for its eigenvalues. Thus any projection has 0 and 1 for its eigenvalues. The multiplicity of 0 as an eigenvalue is the nullity of P , while the multiplicity of 1 is the rank of P .
If the characteristic equation has a root r 1 that is repeated k times, then it is clear that y p (x) = c 1 e r 1 x is at least one solution. [1] However, this solution lacks linearly independent solutions from the other k − 1 roots. Since r 1 has multiplicity k, the differential equation can be factored into [1]