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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.
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]
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography.
One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see Chinese remainder theorem § Hermite interpolation. For yet another method, see, [1] which uses contour integration.
This polynomial is further reduced to = + + which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing () and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were ...
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 :
Macaulay's resultant is a polynomial in the coefficients of these n homogeneous polynomials that vanishes if and only if the polynomials have a common non-zero solution in an algebraically closed field containing the coefficients, or, equivalently, if the n hyper surfaces defined by the polynomials have a common zero in the n –1 dimensional ...
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. [1]