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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 ...
Ruffini's rule can be used when one needs the quotient of a polynomial P by a binomial of the form . (When one needs only the remainder, the polynomial remainder theorem provides a simpler method.) A typical example, where one needs the quotient, is the factorization of a polynomial p ( x ) {\displaystyle p(x)} for which one knows a root r :
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.
The Chinese remainder theorem appears as an exercise [16] in Sunzi Suanjing (between the third and fifth centuries). [17] (There is one important step glossed over in Sunzi's solution: [note 4] it is the problem that was later solved by Āryabhaṭa's Kuṭṭaka – see below.)
Counterexamples by Fujiwara and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a non-negative integer. [ 8 ] On the other hand, Birch's theorem shows that if d is any odd natural number, then there is a number N ( d ) such that any form of degree d in more than N ( d ) variables represents ...
In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions.
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). [9] [10]