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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 ...
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomial + +, with variables and , the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient.
For example, if n = 171 × p × q where p < q are very large primes, trial division will quickly produce the factors 3 and 19 but will take p divisions to find the next factor. As a contrasting example, if n is the product of the primes 13729, 1372933, and 18848997161, where 13729 × 1372933 = 18848997157, Fermat's factorization method will ...
In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors.For example, 21 is the product of 3 and 7 (the result of multiplication), and (+) is the product of and (+) (indicating that the two factors should be multiplied together).
For example, 999 = 3 3 ×37, 1000 = 2 3 ×5 3, 1001 = 7×11×13. Factors p 0 = 1 may be inserted without changing the value of n (for example, 1000 = 2 3 ×3 0 ×5 3). In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers, as
The question of when this happens is rather subtle: for example, for the localization of k[x, y, z]/(x 2 + y 3 + z 5) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x 2 + y 3 + z 7) at the prime ideal (x, y, z) the local ring is a UFD but ...
For example, the number of irreducible factors of a polynomial is the nullity of its Ruppert matrix. [7] Thus the multiplicities m 1 , … , m k {\displaystyle m_{1},\ldots ,m_{k}} can be identified by square-free factorization via numerical GCD computation and rank-revealing on Ruppert matrices.
The entry 4+2i = −i(1+i) 2 (2+i), for example, could also be written as 4+2i= (1+i) 2 (1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right complex half plane with absolute value of the real part larger than or equal to the absolute value of the imaginary part.
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