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A root of a polynomial is a zero of the corresponding polynomial function. [1] The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree , and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically ...
In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding
In algebra, the rational root theorem (or rational root test, rational zero theorem, ... Of these, 1, 2, and –3 equate the polynomial to zero, and hence are its ...
Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). [10] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, f(x) = 0, is the x-axis.
Root-finding of polynomials – Algorithms for finding zeros of polynomials; Square-free polynomial – Polynomial with no repeated root; Vieta's formulas – Relating coefficients and roots of a polynomial; Cohn's theorem relating the roots of a self-inversive polynomial with the roots of the reciprocal polynomial of its derivative.
The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative polynomial root-finding method published in 1970 by Michael A. Jenkins and Joseph F. Traub. They gave two variants, one for general polynomials with complex coefficients, commonly known as the "CPOLY" algorithm, and a more complicated variant for the ...
For example, the 4th order Chebyshev polynomial from the example above is +, which by inspection contains no roots of zero. Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of , which by inspection contains two roots at zero, and may be used in applications ...
Suppose the zeroes z 1, z 2, and z 3 of a third-degree polynomial p(z) are non-collinear. There is a unique ellipse inscribed in the triangle with vertices z 1, z 2, z 3 and tangent to the sides at their midpoints: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative p'(z).