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In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them. All factorization algorithms, including the case of multivariate polynomials over the rational numbers, reduce the problem to this case; see polynomial ...
Modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. [3] For this purpose, even for factoring over the rational numbers and number fields , a fundamental step is a factorization of a polynomial over a finite field .
In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials.
For example, finding a substitution = + + for a cubic equation of degree =, = + + + such that substituting = yields a new equation ′ = + ′ + ′ + ′ such that ′ =, ′ =, or both. More generally, it may be defined conveniently by means of field theory , as the transformation on minimal polynomials implied by a different choice of ...
The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra ), it follows that every polynomial with real coefficients can be factored into ...
For instance, the polynomial x 2 + 3x + 2 is an example of this type of trinomial with n = 1. The solution a 1 = −2 and a 2 = −1 of the above system gives the trinomial factorization: x 2 + 3x + 2 = (x + a 1)(x + a 2) = (x + 2)(x + 1). The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.
The result R = 0 occurs if and only if the polynomial A has B as a factor. Thus long division is a means for testing whether one polynomial has another as a factor, and, if it does, for factoring it out. For example, if a root r of A is known, it can be factored out by dividing A by (x – r).
p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n. The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials.
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