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Alternatively, Horner's method also refers to a method for approximating the roots of polynomials, described by Horner in 1819. It is a variant of the Newton–Raphson method made more efficient for hand calculation by the application of Horner's rule. It was widely used until computers came into general use around 1970.
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials [ 1] —hence the method may be referred to as the FOIL method. The word FOIL is an acronym for the four terms of the product: The general form is. Note that a is both a "first" term and an "outer" term; b is both a "last" and "inner" term, and so forth.
The cyclic redundancy check (CRC) is based on division in the ring of polynomials over the finite field GF (2) (the integers modulo 2 ), that is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around. Any string of bits can be interpreted as the coefficients of a message polynomial of this ...
All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant.
Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string, with a fixed number of zeroes appended, by the "generator polynomial" string except that exclusive or operations replace subtractions.
Cubic equation. Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis at y = 0 ). The case shown has two critical points. Here the function is and therefore the three real roots are 2, -1 and -4. In algebra, a cubic equation in one variable is an equation of the form in which a is not zero.
By multiplying a by every number in the field until the product is one. This is a brute-force search. Since the nonzero elements of GF(p n) form a finite group with respect to multiplication, a p n −1 = 1 (for a ≠ 0), thus the inverse of a is a p n −2. By using the extended Euclidean algorithm.
GF (2) (also denoted , Z/2Z or ) is the finite field with two elements [ 1] (GF is the initialism of Galois field, another name for finite fields). Notations Z2 and may be encountered although they can be confused with the notation of 2 -adic integers . GF (2) is the field with the smallest possible number of elements, and is unique if the ...