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In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
A root (or zero) of a chromatic polynomial, called a “chromatic root”, is a value x where (,) =. Chromatic roots have been very well studied, in fact, Birkhoff’s original motivation for defining the chromatic polynomial was to show that for planar graphs, P ( G , x ) > 0 {\displaystyle P(G,x)>0} for x ≥ 4.
Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equation x 5 − 5 x 4 + 30 x 3 − 50 x 2 + 55 x − 21 = 0 , for which the only real solution is
An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
A root of a nonzero univariate polynomial P is a value a of x such that P(a) = 0. In other words, a root of P is a solution of the polynomial equation P(x) = 0 or a zero of the polynomial function defined by P. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.
If the polynomial has rational roots, for example x 2 − 4x + 4 = (x − 2) 2, or x 2 − 3x + 2 = (x − 2)(x − 1), then the Galois group is trivial; that is, it contains only the identity permutation. In this example, if A = 2 and B = 1 then A − B = 1 is no longer true when A and B are swapped.
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R.Then, the quotients / belong to the field of fractions of R (and possibly are in R itself if happens to be invertible in R) and the roots are taken in an algebraically closed extension.
If the characteristic equation has a root r 1 that is repeated k times, then it is clear that y p (x) = c 1 e r 1 x is at least one solution. [1] However, this solution lacks linearly independent solutions from the other k − 1 roots. Since r 1 has multiplicity k, the differential equation can be factored into [1]