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Naturally, in the case of analytic functions one can estimate the remainder term () by the tail of the sequence of the derivatives f′(a) at the center of the expansion, but using complex analysis also another possibility arises, which is described below.
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 by of a polynomial in of degree 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 ...
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]
Theorem — For any table of nodes there is a continuous function f(x) on an interval [a, b] for which the sequence of interpolating polynomials diverges on [a,b]. [ 14 ] The proof essentially uses the lower bound estimation of the Lebesgue constant, which we defined above to be the operator norm of X n (where X n is the projection operator on ...
The linear approximation is obtained by dropping the remainder: () + ′ (). This is a good approximation when x {\displaystyle x} is close enough to a {\displaystyle a} ; since a curve, when closely observed, will begin to resemble a straight line.
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This construction is analogous to the Chinese remainder theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears. Furthermore, when the order is large, Fast Fourier transformation can be used to solve for the coefficients of the interpolated polynomial.