Factorization
Factorization is a mathematical process that involves breaking down complex numbers or expressions into simpler parts known as factors. In mathematics, this typically means finding the numbers that, when multiplied together, yield the original value. For instance, the factorization of 15 results in the factors 3 and 5. A significant goal of factorization is to reduce numbers to their prime factors—those that are only divisible by 1 and themselves. The fundamental theorem of arithmetic underpins this process, stating that every positive integer greater than 1 has a unique prime factorization. Factorization is not limited to integers; it can also be applied to polynomials, allowing mathematicians to simplify expressions like \(a^2 + ba + c\) into simpler terms. Various techniques exist for factorizing polynomials, including the identification of the highest common factor and other methods, although some require familiarity with specific classes of polynomials. With advancements in technology, sophisticated algorithms have been developed to expedite the factorization process, especially for larger integers used in computer encryption.
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Factorization
Factorization is a process which seeks to take something complex and break it down into smaller, less complex parts. In the realm of mathematics, this usually involves analyzing a number to see what numbers, when multiplied together, produce the original quantity. These numbers, which can be multiplied to produce the original number, are known as factors, and the process of identifying them is referred to as factorization. For example, performing factorization on the value 15 would identify factors of 3 and 5. Often the goal of factoring is to reduce a number to its prime factors, meaning those factors which are only divisible by 1 and by themselves.
Factoring 15 is relatively easy, but as integers become larger the number of calculations required to identify all the factors grows correspondingly difficult. Many computer encryption algorithms that are used to make online information secure use large integer factorization as the lock behind which secrets are stored, because it is so time and resource intensive to determine the factors. Some of these factorization problems are so complex that it would take a computer, doing nothing else but calculating factors, many years to complete its work.
Overview
Factorization is the basis of the fundamental theorem of arithmetic, which states that there is a unique factorization consisting only of prime numbers for every positive integer that is greater than 1. For example, for the number 1,400, factorization identifies the prime factors that, when multiplied together, produce this value as 2 × × 2 × 2 × 5 × 5 × 7. This means that multiplying these numbers together produces a result of 1,400 and that no other combination of prime factors will produce this same result. For this reason, the fundamental theorem of arithmetic can also be referred to as the unique prime factorization theorem or as simply the unique factorization theorem.
In addition to being performed on integers, factorization can also be used with polynomials. An example of this would be taking the polynomial expression a2 +ba + c and attempting to discover what simpler terms and expressions the polynomial could be broken down into. There are several different techniques that can be used to factor polynomials. Some of the newer processes involve the use of computers that have been programmed to perform sophisticated operations quickly. There are also methods for making the calculations by hand, as has been studied for hundreds of years, but many of these are limited in the range of polynomials they can factor, because they rely, at least in part, on the mathematician being able to recognize that a particular polynomial belongs to a certain class that is amenable to factorization in a particular way. One classic technique is called the highest common factor, because it requires the identification of the highest factor common to all of the terms of the polynomial and then factoring it out by employing the distributive property. Other approaches include the use of the factor theorem or factoring by the grouping method, though this latter technique is not always effective.
Bibliography
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Francisco, Christopher. Closures, Finiteness and Factorization. Berlin: De Gruyter, 2012.
Griffiths, Martin. "Thematic Mathematics: The Combinatorics of Prime Factorizations." Teaching Mathematics and Its Applications 29.1 (2010): 25-40.
Rego, Drumond L. Factorization Models for Multi-Relational Data. Göttingen: Cuvillier, 2014.
Riesel, Hans. Prime Numbers and Computer Methods for Factorization. Boston: Birkhäser, 2012.
Wheater, Carolyn C. Basic Math. New York: McGraw, 2012.