Search results
Results from the WOW.Com Content Network
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 ...
Given a positive integer n, Fermat's factorization method relies on finding numbers x and y satisfying the equality = We can then factor n = x 2 − y 2 = (x + y)(x − y). This algorithm is slow in practice because we need to search many such numbers, and only a few satisfy the equation.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b). In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
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:
Algorithm: SFF (Square-Free Factorization) Input: A monic polynomial f in F q [x] where q = p m Output: Square-free factorization of f R ← 1 # Make w be the product (without multiplicity) of all factors of f that have # multiplicity not divisible by p c ← gcd(f, f′) w ← f/c # Step 1: Identify all factors in w i ← 1 while w ≠ 1 do y ...
Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all the prime factors of n congruent to 3 modulo 4 occur to an even ...
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 ...
A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials f = a 1 a 2 2 a 3 3 ⋯ a n n = ∏ k = 1 n a k k {\displaystyle f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{n}^{n}=\prod _{k=1}^{n}a_{k}^{k}\,}