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The rings for which such a theorem exists are called Euclidean domains, but in this generality, uniqueness of the quotient and remainder is not guaranteed. [ 8 ] Polynomial division leads to a result known as the polynomial remainder theorem : If a polynomial f ( x ) is divided by x − k , the remainder is the constant r = f ( k ).
In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) [1] is an application of Euclidean division of polynomials.It states that, for every number , any polynomial is the sum of () and the product of and a polynomial in of degree one less than the degree of .
Remainder theorem may refer to: Polynomial remainder theorem; Chinese remainder theorem This page was last edited on 29 December 2019, at 22:03 (UTC). Text is ...
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1).
The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division. [19] When the denominator b(x) is monic and linear, that is, b(x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a(x) by b(x) is the evaluation a ...
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Firstly, their definition through determinants allows bounding, through Hadamard inequality, the size of the coefficients of the GCD. Secondly, this bound and the property of good specialization allow computing the GCD of two polynomials with integer coefficients through modular computation and Chinese remainder theorem (see below).
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