Factorization

This article is about the mathematical concept. For other uses, see factor. FACTOR may also refer to the Object Oriented programming requirements caputre acronym "Functionality Application domain Conditions Technology Objects and Responsibility"

In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and British English or UK English ( BrE, BE, en-GB) is the broad term used to distinguish the forms of the English language used in the A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a product is the Result of multiplying, or an expression that identifies factors to be multiplied For example, the number 15 factors into primes as 3 × 5, and the polynomial x² − 4 factors as (x − 2)(x + 2). In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In all cases, a product of simpler objects is obtained.

The aim of factoring is usually to reduce something to "basic building blocks," such as numbers to prime numbers, or polynomials to irreducible polynomials. In Mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given set Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Mathematics and Computer algebra, Polynomial Factorization typically refers to factoring a polynomial into Irreducible polynomials In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at

The opposite of factorization is expansion. In Mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition This is the process of multiplying together factors to recreate the original, "expanded" polynomial. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA. Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key In Cryptography, RSA is an Algorithm for Public-key cryptography.

A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* In the mathematical discipline of Linear algebra, a triangular matrix is a special kind of Square matrix where the entries either below or above the There are different types: QR decomposition, LQ, QL, RQ, RZ. In Linear algebra, the QR decomposition (also called the QR factorization) of a matrix is a decomposition of the matrix into an orthogonal

Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

1 Prime factorization of an integer
2 Factoring a quadratic polynomial
3 Factoring other polynomials
- 3.1 Sum/difference of two cubes
- 3.2 Sum/difference of any two numbers raised to the same power
4 Factoring by grouping
5 Other common formulas
6 Factoring in mathematical logic
7 See also
8 External links

Prime factorization of an integer

Main article: Integer factorization

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 For very large numbers, no efficient algorithm is known. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation For smaller numbers, however, there are a variety of different algorithms that can be applied.

Factoring a quadratic polynomial

Any quadratic polynomial over the complex numbers (polynomials of the form $a x 2 + b x + c$ where $a$ , $b$ , and $c$ ∈ $\mathbb{C}$ ) can be factored into an expression with the form $a(x - \alpha)(x - \beta) \$ using the quadratic formula. In mathematics a quadratic polynomial is a Polynomial whose degree is 2 Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In mathematics the word expression is a term for any well-formed combination of mathematical symbols In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. The method is as follows:

$ax^2 + bx + c = a(x - \alpha)(x - \beta) = a\left(x - \left(\frac{-b + \sqrt{b^2-4ac}}{2a}\right)\right) \left(x - \left(\frac{-b - \sqrt{b^2-4ac}}{2a}\right)\right)$

where $α$ and $β$ are the two roots of the polynomial, found with the quadratic formula. This article is about the zeros of a function which should not be confused with the value at zero. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree.

Polynomials factorable over the integers

Quadratic polynomials can sometimes be factored into two binomials with simple integer coefficients, without the need to use the quadratic formula. In a quadratic equation, this will expose its two roots. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. The formula

$ax^2+bx+c \,\!$

would be factored into:

$(mx+p)(nx+q) \,\!$

where

$mn = a, \,$

$pq = c, \mbox{ and} \,$

$pn + mq = b. \,$

You can then set each binomial equal to zero, and solve for x to reveal the two roots. Factoring does not involve any other formulas, and is mostly just something you see when you come upon a quadratic equation.

Take for example 2x² − 5x + 2 = 0. Because a = 2 and mn = a, mn = 2, which means that of m and n, one is 1 and the other is 2. Now we have (2x + p)(x + q) = 0. Because c = 2 and pq = c, pq = 2, which means that of p and q, one is 1 and the other is 2 or one is −1 and the other is −2. A guess and check of substituting the 1 and 2, and −1 and −2, into p and q (while applying pn + mq = b) tells us that 2x² − 5x + 2 = 0 factors into (2x − 1)(x − 2) = 0, giving us the roots x = {0. 5, 2}

If a polynomial with integer coefficients has a discriminant that is a perfect square, that polynomial is factorable over the integers.

For example, look at the polynomial 2x² + 2x - 12. If you substitute the values of the expression into the quadratic formula, the discriminant $b 2 - 4 a c$ becomes 2² - 4 × 2 × -12, which equals 100. 100 is a perfect square, so the polynomial 2x² + 2x - 12 is factorable over the integers; its factors are 2, (x - 2), and (x + 3).

Now look at the polynomial x² + 93x - 2. Its discriminant, 93² - 4 × 1 × -2, is equal to 8657, which is not a perfect square. So x² + 93x - 2 cannot be factored over the integers.

Perfect square trinomials

A visual proof of the identity (a+b)²=a²+2ab+b²

Some quadratics can be factored into two identical binomials. These quadratics are called perfect square trinomials. Perfect square trinomials can be factored as follows:

$a^2 + 2ab + b^2 = (a + b)^2\,\!$

$a^2 - 2ab + b^2 = (a - b)^2\,\!$

Sum/difference of two squares

Main article: Difference of two squares

Another common type of algebraic factoring is called the difference of two squares. In Mathematics, the difference of two squares is when a number is squared, or multiplied by itself and is then subtracted from another squared number In Mathematics, the difference of two squares is when a number is squared, or multiplied by itself and is then subtracted from another squared number It is the application of the formula

$a^2 - b^2 = (a+b)(a-b) \,\!$

to any two terms, whether or not they are perfect squares. If the two terms are subtracted, simply apply the formula. If they are added, the two binomials obtained from the factoring will each have an imaginary term. This formula can be represented as

$a^2 + b^2 = (a+bi)(a-bi) \,\!$ .

For example, $4 x 2 + 49$ can be factored into $(2 x + 7 i)(2 x - 7 i)$ .

Factoring other polynomials

Sum/difference of two cubes

Another less-used but still common formula for factoring is the sum or difference of two cubes. The sum can be represented by

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)\,\!$

and the difference by

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)\,\!$

For example, x³ − 10³ (or x³ − 1000) can be factored into (x − 10)(x² + 10x + 100).

Sum/difference of any two numbers raised to the same power

In general, $(a - b)$ is a factor of $a n - b n$ where $n$ is a positive integer. So,

$a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + ... + a^2b^{n-3} + ab^{n-2} + b^{n-1}) \,\!$

Also, $(a + b)$ is a factor of $a n - b n$ where $n$ is a positive even integer. Such that,

$a^n - b^n = (a + b)(a^{n-1} - a^{n-2}b + a^{n-3}b^2 - ... - a^2b^{n-3} + ab^{n-2} - b^{n-1}) \,\!$

Likewise, $(a + b)$ is a factor of $a n + b n$ where $n$ is a positive odd integer. So that,

$a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + a^{n-3}b^2 - ... + a^2b^{n-3} - ab^{n-2} + b^{n-1}) \,\!$

Factoring by grouping

Another way to factor some equations is factoring by grouping. This is done by placing the terms in an expression into two or more groups, where each group can be factored by a known method. The results of these factorizations can sometimes be combined to make an even more simplified expression.

For example, suppose you had the expression

$4x^3\sin^2x-312x^2\sin^2x+4620x\sin^2x-8024\sin^2x-3x^3+234x^2-3465x+6018 \,$

which upon first glance looks like an unwieldy expression. One logical step, if you decide to factor by grouping, would be to combine all of the expressions with $\sin x\,\!$ and all without $\sin x\,\!$ . Then you would have the expression

$(4x^3\sin^2x-312x^2\sin^2x+4620x\sin^2x-8024\sin^2x)-(3x^3-234x^2+3465x-6018) \,$

where each of the two groups can be factored giving us

$4\sin^2x(x^3-78x^2+1155x-2006) - 3(x^3-78x^2+1155x-2006) \,$

This can be further simplified into

$(4\sin^2x -3)(x^3-78x^2+1155x-2006) \,$

when can then be factored into

$(4\sin^2x -3)(x-59)(x-17)(x-2) \,$

and finally

$(2\sin x+\sqrt 3)(2\sin x-\sqrt 3)(x-59)(x-17)(x-2) \,$

which is the expression in fully factored form.

Other common formulas

There are many additional formulas that can be used to easily factor a polynomial. Some common ones are listed below.

Expanded form	Factored form
$a^3+b^3+c^3-3abc\,\!$	$(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\,\!$
$a^2(b+c)+b^2(c+a)+c^2(a+b)+2abc\,\!$	$(a+b)(b+c)(c+a)\,\!$
$(a+b)(b+c)(c+a)+abc\,\!$	$(a+b+c)(ab+bc+ca)\,\!$
$bc(b-c)+ca(c-a)+ab(a-b)\,\!$	$-(a-b)(b-c)(c-a)\,\!$
$a^2(b+c)+b^2(c+a)+c^2(a+b)+3abc\,\!$	$(a+b+c)(ab+bc+ca)\,\!$
$a^2(b-c)+b^2(c-a)+c^2(a-b)\,\!$	$-(a-b)(b-c)(c-a)\,\!$
$a^3(b-c)+b^3(c-a)+c^3(a-b)\,\!$	$-(a-b)(b-c)(c-a)(a+b+c)\,\!$
$a^4 + 4b^4 \,\!$ (Sophie Germain's identity)	$(a^2 + 2ab + 2b^2) (a^2 - 2ab + 2b^2) \,\!$

Factoring in mathematical logic

In mathematical logic and automated theorem proving, factoring is the technique of deriving a single, more specific atom from a disjunction of two more general unifiable atoms. This article is about the mathematician Marie-Sophie Germain See also Sophie Germain primes Marie-Sophie Germain ( April 1, 1776 Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Automated theorem proving ( ATP) or automated deduction, currently the most well-developed subfield of Automated reasoning (AR is the History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Mathematical logic, in particular as applied to Computer science, a unification of two terms is a join (in the lattice sense with respect For example, from ∀ X, Y : P(X, a) or P(b, Y) we can derive P(b, a).

External links

A page about factorization, Algebra, Factoring
WIMS Factoris is an online factorization tool. Program synthesis comprises a range of technologies for the automatic generation of executable computer programs from high-level Specifications of their behaviour In the mathematical discipline of Linear algebra, a matrix decomposition is a Factorization of a matrix into some Canonical form. In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written In Mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the The FOIL rule, also sometimes known as the double distributive property or more colloquially as foiling, is commonly taught to U
Polynomial Factoring is a comprehensive tutorial resource on basic factoring of polynomials.

Dictionary

-noun

(mathematics): The process of creating a list of factors.
(mathematics): An expression listing items that, when multiplied together, will produce a desired quantity.
A list of factors.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents