mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field...

mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition...

integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful within number...

factoring polynomials over the integers and over the rational numbers use this result (see Factorization of polynomials). The irreducibility of a polynomial over...

theorem about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property...

{x+y}{2}}\right)^{2}-\left({\frac {x-y}{2}}\right)^{2}.} The proof of the factorization identity is straightforward. Starting from the right-hand side, apply...

integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer...

square-free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity of the original polynomial. The greatest...

many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings...

part–content factorization (see Factorization of polynomials § Primitive part–content factorization). Then the factorization problem is reduced to factorize separately...

In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every...

test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials). These algorithms are not practicable...

cyclotomic polynomials are irreducible polynomials over the integers, such a factorization cannot come from an algebraic factorization of the polynomial. Nevertheless...

factorization be solved in polynomial time on a classical computer? More unsolved problems in computer science In mathematics, integer factorization is...

roots of polynomials is a long-standing problem that has been extensively studied throughout the history and substantially influenced the development of mathematics...

has somewhat different properties: finding the form requires factorization of polynomials, and as a consequence the primary rational canonical form may...

p(x) is a polynomial, and, for each j, the denominator gj (x) is a power of an irreducible polynomial (i.e. not factorizable into polynomials of positive...

factor Factorization Euler's factorization method Integer factorization Program synthesis Table of Gaussian integer factorizations Unique factorization Lehman...

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...

Spectral Factorization has been used to find determinantal matrix representations for bivariate stable polynomials and real zero polynomials. Given a...

factorization of polynomials Fundamental theorem of arithmetic, a theorem regarding prime factorization Fundamental analysis, the process of reviewing and...

arithmetic of the integers, and the arithmetic of the univariate polynomials, as polynomial-time algorithms are known for square-free factorization of polynomials...

for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations...

division, the radical of an ideal, the ideal of zero-dimensional schemes, Poincaré series and Hilbert functions, factorization of polynomials, and toric ideals...

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}...

polynomial. As with the cyclotomic polynomials more generally, Φ 4 {\displaystyle \Phi _{4}} is an irreducible polynomial, so this factorization of infinitely...

examples. Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as:...

algorithm, named after Sergei Evdokimov, is an algorithm for factorization of polynomials over finite fields. It was the fastest algorithm known for this...

( x ) {\displaystyle \Phi _{n}(x)} , provided integer factorization and division of polynomials are available. Many computer algebra systems, such as...

factormod() function (formerly factorcantor()). Polynomial factorization Factorization of polynomials over finite fields Cantor, David G.; Zassenhaus...