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If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial ...
The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 (sequence A046760 in the OEIS). An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.
A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, [10] has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.
Exceptions are made if the unit is commonly known by another name (for example, 1 micron = 10 −6 metre). Within each table, the units are listed alphabetically, and the SI units (base or derived) are highlighted.
In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f ( x ) {\displaystyle f(x)} is a polynomial, then x − a {\displaystyle x-a} is a factor of f ( x ) {\displaystyle f(x)} if and only if f ( a ) = 0 {\displaystyle f(a)=0} (that is, a {\displaystyle a} is a root of the polynomial).
Unit-Scale-Invariant Singular-Value Decomposition: =, where S is a unique nonnegative diagonal matrix of scale-invariant singular values, U and V are unitary matrices, is the conjugate transpose of V, and positive diagonal matrices D and E.
List of orders of magnitude for area 10 −70 to 10 −9 square metres; Factor (m 2) Multiple Value Item 10 −70 2.6 × 10 −70 m 2: Planck area, [1] 10 −60: 1 square quectometre: 10 −54: 1 square rontometre: 10 −52 100 rm 2: 1 shed [2] 10 −48: 1 square yoctometre (ym 2) 1 ym 2 10 −43 100,000 ym 2: 1 femtobarn [3] 10 −42