A factor, also known as a divisor, is a number or variable that divides into another number or algebraic expression without leaving a remainder. For example, the factors of 28 are 1, 2, 4, 7, 14, and 28. Although it is true that 28 is also divisible by the negative of each of these, the term "factors" is usually taken to mean only the positive divisors. See also highest common factor.

 

Factorization, or factoring, is the decomposition of an object into a product of factors. For example, the number 15 factorizes into prime numbers as 3 × 5; and the polynomial x ² – 4 factorizes as (x – 2)(x + 2). The aim of factoring is usually to reduce something to basic building blocks, such as numbers to prime numbers, or polynomials to linear expressions. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra (see also below). Integer factorization for large integers appears to be a difficult problem; there are no known methods for solving it quickly and, for this reason, it has formed the basis of some public key cryptography algorithms.

 

Factors of a polynomial

The factors of a polynomial are generally found by a mixture of guesswork and rules of thumb. This is helped by certain standard results:

 

    x ² - y ² = (x + y)(x – y)
    x ³ – y ³ = (x – y)(x ² + xy + y ²)
    x ³ + y ³ = (x + y)(x ² – xy + y ²)
    x ² + 2xy + y ² = (x + y) ².

 

Moreover, to find the factors of a polynomial of the form x ² + bx + c we know that, if the factors are (x + p) and (x + q), p + q = b and p.q = c. Hence x ² – 3x + 2 has factors (x – 2) and (x – 1), since (–2) + (–1) = (–3) and (–2)(–1) = 2.