What is factoring?

What Does factoring Mean

Factoring is a term used in mathematics to refer to the act and result of factoring . This verb ( factoring ), meanwhile, refers to the decomposition of a polynomial into the product of other polynomials of a lower degree or to the expression of an integer from the product of its divisions.

It can be said that factoring allows decomposing an algebraic expression into factors to present it in a simpler way . It should be noted that the factors are expressions that are subjected to multiplication to obtain a product.
Take the case of whole number factorization . This process involves the decomposition of composite numbers into divisors that, when multiplied, make it possible to obtain the number in question.

According to the single factorization theorem , also known as the fundamental theorem of arithmetic , a positive integer can only be decomposed one way into prime numbers. On the other hand, a prime number is called the natural number that is greater than 1 and that only has two natural divisors: 1 and itself .
Let's look at the case of number 81 :




Factoring 81 into prime numbers, thus, is 3 to the power of 4 (3 x 3 x 3 x 3).
Returning to the definition of the fundamental theorem of arithmetic, we must understand that it applies to all integers greater than 1, that is, positive. Point out that in this group we can only find prime numbers or unique products of prime numbers, that is, this second possibility is fixed for each case. Since in the case of multiplication we have the commutative property, according to which the order of the factors does not affect the product, we can alter the sequence of the prime numbers resulting from the factorization.
One can also speak of factorization of polynomials . In this case, polynomials are factored by appealing to coefficients in a certain field or domain. These calculations are usually carried out with computer algebra systems. The matrix factorization finally relates to the decomposition of a matrix as the product of two matrices to the least.
Let's take a closer look at some of the concepts expressed in the previous paragraph. A field , in this context, is an algebraic system in which addition and multiplication operations can be carried out respecting the commutative, distributive and associative properties of the second with respect to the first. It also admits the additive inverse , the multiplicative inverse and two neutral elements that open the doors to subtraction and division (the latter cannot be done by zero).

With regard to the computer algebra system , for which this type of factorization represents one of the most important tools, it is a program executed by a processor that allows calculations to be carried out symbolically. It differs from a traditional calculator in that it allows formulas and equations to be solved symbolically rather than numerically. This means that you can interpret variables as such instead of just accepting numbers.
Polynomial factoring has a long history. It dates back to 1793, when the scientist Hermann Schubert made the first description of an algorithm designed for this purpose. Almost a century later, in 1882, Leopold Kronecker continued to work on Schubert's proposal and expanded it to include multivariate and coefficient polynomials.
Despite all this, the greatest volume of discoveries and theories around this type of factorization emerged in the second half of the 20th century. Broadly speaking, we can mention two groups of methods to calculate polynomial factorization: the classical ones (obtaining linear factors and the Kronecker method); the modern ones (the LLL algorithm and Trager's method).

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