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  2. Primitive root modulo n - Wikipedia

    en.wikipedia.org/wiki/Primitive_root_modulo_n

    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.

  3. Chromatic polynomial - Wikipedia

    en.wikipedia.org/wiki/Chromatic_polynomial

    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.

  4. Quintic function - Wikipedia

    en.wikipedia.org/wiki/Quintic_function

    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

  5. Newton's method - Wikipedia

    en.wikipedia.org/wiki/Newton's_method

    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.

  6. Polynomial - Wikipedia

    en.wikipedia.org/wiki/Polynomial

    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.

  7. Galois theory - Wikipedia

    en.wikipedia.org/wiki/Galois_theory

    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.

  8. Vieta's formulas - Wikipedia

    en.wikipedia.org/wiki/Vieta's_formulas

    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.

  9. Characteristic equation (calculus) - Wikipedia

    en.wikipedia.org/wiki/Characteristic_equation...

    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]