Search results
Results from the WOW.Com Content Network
is a factorization into content and primitive part. Every polynomial q with rational coefficients may be written =, where p ∈ Z[X] and c ∈ Z: it suffices to take for c a multiple of all denominators of the coefficients of q (for example their product) and p = cq. The content of q is defined as:
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any can be trivially written as () (/) whenever is not zero.
For the fourth time through the loop we get y = 1, z = x + 2, R = (x + 1)(x + 2) 4, with updates i = 5, w = 1 and c = x 6 + 1. Since w = 1, we exit the while loop. Since c ≠ 1, it must be a perfect cube. The cube root of c, obtained by replacing x 3 by x is x 2 + 1, and calling the
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite.The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...
In general, not every GCD in the above product will be a non-trivial factor of (), but some are, providing the factors we seek. Berlekamp's algorithm finds polynomials g ( x ) {\displaystyle g(x)} suitable for use with the above result by computing a basis for the Berlekamp subalgebra.
A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that: E and M both contain all isomorphisms of C and are closed under composition. Every morphism f of C can be factored as f = m ∘ e {\displaystyle f=m\circ e} for some morphisms e ∈ E {\displaystyle e\in E} and m ∈ M {\displaystyle m\in ...