nonfactor

Nonfactor is an informal mathematical term used to describe an object that is not a factor (divisor) of another object. The phrase is not standardized, and its precise meaning can vary by context; most mathematicians prefer explicit phrases such as “does not divide” or “is not a factor.”

In elementary number theory, a nonfactor of a positive integer n is any integer d that does not divide n, meaning there is no integer q with n = d·q. For example, with n = 12, the numbers 5, 7, 8, 9, 10, and 11 are nonfactors of 12, while 1, 2, 3, 4, 6, and 12 are factors. Nonfactors can be negative as well: −5 is a nonfactor of 12 since 12 is not divisible by −5.

In algebra, a nonfactor of a polynomial f(x) over a given ring is a polynomial g(x) in that ring such that f(x) is not divisible by g(x). For example, in the ring of real-coefficient polynomials, f(x) = x^2 + 1 has no real linear factor, so any real linear polynomial such as x − r with r real is a nonfactor of f(x). More generally, a nonfactor is simply a polynomial that does not divide f(x).

Limitations and terminology: the term nonfactor is rarely used in formal writing; standard terms are “does not divide” or “not a factor.” Outside strict mathematics, the word may appear in branding or informal discourse without a precise technical meaning.

See also

- Factor (mathematics)

- Divisor

- Polynomial factorization

- Nondivisor

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