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

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

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

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...

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

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

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

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

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...

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

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

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

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...

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

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

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

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

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...

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

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

the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states...

algorithm produces a list of codewords (it is a list-decoding algorithm) and is based on interpolation and factorization of polynomials over GF(2m) and its...

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

simplification of expressions, differentiation using the chain rule, polynomial factorization, indefinite integration, etc. Computer algebra is widely used to...

Its factorization was announced on April 1, 1991, by Arjen K. Lenstra. Reportedly, the factorization took a few days using the multiple-polynomial quadratic...

unique factorization domains ⊃ principal ideal domains ⊃ euclidean domains ⊃ fields ⊃ algebraically closed fields Formally, a unique factorization domain...