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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.
A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or p-adic) root α of the polynomial () to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α 2, α 3, . . . with integer coefficients, which might be an ...
Because (a + 1) 2 = a, a + 1 is the unique solution of the quadratic equation x 2 + a = 0. On the other hand, the polynomial x 2 + ax + 1 is irreducible over F 4, but it splits over F 16, where it has the two roots ab and ab + a, where b is a root of x 2 + x + a in F 16. This is a special case of Artin–Schreier theory.
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 ...
Therefore, the graph of the function f(x − h) = (x − h) 2 is a parabola shifted to the right by h whose vertex is at (h, 0), as shown in the top figure. In contrast, the graph of the function f(x) + k = x 2 + k is a parabola shifted upward by k whose vertex is at (0, k), as shown in the center figure.
In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f ( x ) {\displaystyle f(x)} is a polynomial, then x − a {\displaystyle x-a} is a factor of f ( x ) {\displaystyle f(x)} if and only if f ( a ) = 0 {\displaystyle f(a)=0} (that is, a {\displaystyle a} is a root of the polynomial).
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In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial 3x 2 − 5x + 4 is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2.