Factorization

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This article is about the mathematical concept. For the financial term see factoring (trade).

In mathematics, factorization 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. For example, the number 15 factors into primes as 3 × 5; and the polynomial x² − 4 factors as (x − 2)(x + 2). In all cases, we obtain a product of simpler things.

The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.

The opposite of factorization is expansion. This is the process of multiplying together factors to recreate the original, "expanded" polynomial.

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.

A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.

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.

Contents 1 Factoring a quadratic polynomial 1.1 Perfect square trinomials 2 Sum/difference of two squares 3 Sum/difference of two cubes 4 Prime factorization of an integer 5 Factoring in mathematical logic 6 See also 7 External links

Factoring a quadratic polynomial

Quadratic equations (equations in the form of <math>ax^2+bx+c=0</math>) can sometimes be factored into two binomials with simple integer coefficients, thus exposing their roots, without the need to use the quadratic formula. The formula

<math>ax^2+bx+c=0 \,\!</math>

would be factored into

<math>(mx+p)(nx+q)=0 \,\!</math>

where

mn = a,

pq = c, 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 = {¹⁄₂, 2}

Perfect square trinomials

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

<math> a^2 + 2ab + b^2 = (a + b)^2\,\!</math>

<math> a^2 - 2ab + b^2 = (a - b)^2\,\!</math>

Sum/difference of two squares

Template:Main Another common type of algebraic factoring is called the difference of two squares. It is the application of the formula

<math> a^2 - b^2 = (a+b)(a-b) \,\!</math>

to any two terms, whether or not they are perfect squares. If the two terms are subtraced, 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

<math> a^2 + b^2 = (a+bi)(a-bi) \,\!</math>

For example, <math>4x + 49</math> can be factored into <math>(2\sqrt {x} + 7i)(2\sqrt {x} - 7i)</math>.

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

<math> a^3 + b^3 = (a + b)(a^2 - ab + b^2)\,\!</math>

and the difference by

<math> a^3 - b^3 = (a - b)(a^2 + ab + b^2)\,\!</math>

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

In general, the difference of any two numbers raised to the same (positive integer) power

<math> a^n - b^n\,\!</math>

contains the factor (a - b), and the sum of any two numbers raised to the same (positive integer) odd power

<math> a^n + b^n\,\!</math>

and the difference of any two numbers raised to the same (positive integer) even power contain the factor (a + b).

Prime factorization of an integer

Template:Main By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm. For very large numbers, however, no efficient algorithm is known. For smaller numbers, however, there are a variety of different algorithms that can be applied. (See prime factorization algorithms)

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. For example, from ∀ X, Y : P(X, a) or P(b, Y) we can derive P(b, a).

See also

• Integer factorization
• Prime factorization algorithm
• Program synthesis
• Unique factorization
• Polynomial expansion, the opposite of factorization

External links

• A page about factorization, Algebra, Factoring
• WIMS Factoris is an online factorization tool.
• Polynomial Factoring is a comprehensive tutorial resource on basic factoring of polynomials.da:Faktorisering

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