Factoring is a term that is used in the field of math to allude to act and result of factoring. This verb (factorize), meanwhile, refers to the decomposition of a polynomial in the product of other polynomials of lower degree or to the expression of an integer from the product of its divisions.
It can be said that factoring allows decompose an algebraic expression into factors to present it in a simpler way. It should be noted that factors They are expressions that are subjected to multiplication to obtain a product.
Take the case of whole number factoring. This process involves decomposition of the composite numbers into divisors that, when multiplied, make it possible to obtain the number in question.
According to unique factorization theorem, also know as The fundamental theorem of arithmetic, a positive integer can only be decomposed one way into prime numbers. Is named Prime number, on the other hand, to the natural number that is greater than 1 and that only has two natural dividers: the 1 Y the same.
Let’s look at the case of the number 81:
The factorization of 81 in prime numbers, thus, is 3 to the power of 4 (3 x 3 x 3 x 3).
Returning to the definition of theorem fundamental 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.
It can also be spoken of factorization of polynomials. In this case, they are factored polynomials appealing to coefficients in a certain field or domain. These calculations are usually carried out with computer algebra systems. The matrix factorizationFinally, it refers to the decomposition of a matrix as the product of at least two matrices.
Let’s take a closer look at some of the concepts expressed in the previous paragraph. A countrysideIn this context, it 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 supports the additive inverse, the Inverse multiplicative and two neutral elements that open the doors to subtraction and division (the latter cannot be done by zero).
With respect to system of computational algebra, for which this type of factorization represents one of the most important tools, it is a program executed by a processor that allows to perform calculations in a symbolic way. 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 takes us back to the year 1793, when the scientist Hermann Schubert made the first description of a algorithm designed for this purpose. Almost a century later, in 1882, Leopold Kronecker He continued working 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 the polynomial factorization: the classics (obtaining linear factors and the Kronecker method); the modern ones (the LLL algorithm and Trager’s method).