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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. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n = 1.
A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or p-adic) root α of the polynomial () to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α 2, α 3, . . . with integer coefficients, which might be an ...
For the largest such that (if any), is called the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial + + is 4. This can be generalised to multivariate polynomials with respect to a monomial order, see Gröbner basis § Leading term, coefficient and monomial.
Two monic polynomials are associated if and only if they are equal, since the multiplication of a polynomial by a nonzero constant produces a polynomial with this constant as its leading coefficient. Divisibility induces a partial order on monic polynomials. This results almost immediately from the preceding properties.
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 factors ...
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.
The Lindsey–Fox algorithm, named after Pat Lindsey and Jim Fox, is a numerical algorithm for finding the roots or zeros of a high-degree polynomial with real coefficients over the complex field. It is particularly designed for random coefficients but also works well on polynomials with coefficients from samples of speech, seismic signals, and ...
All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant.