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Modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. [3] For this purpose, even for factoring over the rational numbers and number fields, a fundamental step is a factorization of a polynomial over a finite field.
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 ...
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.
For instance, the polynomial x 2 + 3x + 2 is an example of this type of trinomial with n = 1. The solution a 1 = −2 and a 2 = −1 of the above system gives the trinomial factorization: x 2 + 3x + 2 = (x + a 1)(x + a 2) = (x + 2)(x + 1). The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.
If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1. Q ( x ) is simply the quotient obtained from the division process; since r is known to be a root of P ( x ), it is known that the remainder must be zero.
In the second step, the distributive law is used to simplify each of the two terms. Note that this process involves a total of three applications of the distributive property. In contrast to the FOIL method, the method using distributivity can be applied easily to products with more terms such as trinomials and higher.
The algorithm is used to factorize a number =, where is a non-trivial factor. A polynomial modulo , called () (e.g., () = (+)), is used to generate a pseudorandom sequence.It is important to note that () must be a polynomial.
A number of results give techniques for locating and testing primitiveness of trinomials. [4] For polynomials over GF(2), where 2 r − 1 is a Mersenne prime, a polynomial of degree r is primitive if and only if it is irreducible. (Given an irreducible polynomial, it is not primitive only if the period of x is a non-trivial factor of 2 r − 1 ...