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Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, (x – a)(x – b) = x 2 – (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.
The following names are assigned to polynomials according to their degree: [2] [3] [4] Special case – zero (see § Degree of the zero polynomial, below) Degree 0 – non-zero constant [5] Degree 1 – linear; Degree 2 – quadratic; Degree 3 – cubic; Degree 4 – quartic (or, if all terms have even degree, biquadratic) Degree 5 – quintic
The class of methods is based on converting the problem of finding polynomial roots to the problem of finding eigenvalues of the companion matrix of the polynomial, [1] in principle, can use any eigenvalue algorithm to find the roots of the polynomial. However, for efficiency reasons one prefers methods that employ the structure of the matrix ...
The Cayley resolvent is a resolvent for the maximal solvable Galois group in degree five. It is a polynomial of degree 6. These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for ...
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
Moreover, if the polynomial degree is a power of 2 and the roots are all real, then if there is a root that can be expressed in real radicals it can be expressed in terms of square roots and no higher-degree roots, as can the other roots, and so the roots are classically constructible. Casus irreducibilis for quintic polynomials is discussed by ...
For polynomials with real or complex coefficients, it is not possible to express a lower bound of the root separation in terms of the degree and the absolute values of the coefficients only, because a small change on a single coefficient transforms a polynomial with multiple roots into a square-free polynomial with a small root separation, and ...