Polynomial Equation

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An Polynomial Equation is a mathematical equation ([math]A = B[/math]) where [math]A[/math] and [math]B[/math] are polynomials with function coefficients in some field.



References

2014

  • (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/algebraic_equation Retrieved:2014-7-26.
    • In mathematics, an algebraic equation or polynomial equation is an equation of the form  :[math]P = Q[/math] where P and Q are polynomials with coefficients in some field, often the field of the rational numbers. For most authors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate and the term polynomial equation is usually preferred to algebraic equation.

      For example,  :[math]x^5-3x+1=0[/math]

      is an algebraic equation with integer coefficients and  :[math]y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}[/math]

      is a multivariate polynomial equation over the rationals.

      Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression with a finite number of operations involving just those coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations but not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of an univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).


  • (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/polynomial#Polynomial_equations Retrieved:2014-7-26.
    • A polynomial equation, also called algebraic equation, is an equation of the form  :[math]a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0 = 0[/math] For example, : [math] 3x^2 + 4x -5 = 0 \,[/math] is a polynomial equation.

      In case of a univariate polynomial equation, the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation stands in contrast to a polynomial identity like {{is the polynomial equation corresponding to . The solutions of this equation are called the roots of the polynomial; they are the zeroes of the function (corresponding to the points where the graph of meets the -axis). A number is a root of if and only if the polynomial (of degree one in ) divides . It may happen that divides more than once: if divides then is called a multiple root of , and otherwise is called a simple root of . If is a nonzero polynomial, there is a highest power such that divides , which is called the multiplicity of the root in . When is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined (or infinite) multiplicity. With this exception made, the number of roots of , even counted with their respective multiplicities, cannot exceed the degree of . The relation between the roots of a polynomial and its coefficients is described by Viète's formulas.

      Some polynomials, such as , do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors , one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.

      There is a difference between approximating roots and finding exact expressions for roots. Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times (see quadratic equation), and for polynomials of degree 3 or 4 similar formulas (using cube roots in addition to square roots) were found in the 16th century (see cubic function and quartic function for the formulas and Niccolò Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta for historical details). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general (finite) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel–Ruffini theorem). In 1830, Évariste Galois, studying the permutations of the roots of a polynomial, extended the Abel–Ruffini theorem by showing that, given a polynomial equation, one may decide if it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and Group theory, two important branches of modern mathematics. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).

      Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the Jenkins–Traub method, Laguerre's method, Durand–Kerner method or by some other root-finding algorithm.

      For polynomials in more than one indeterminate the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero. However for certain sets of such polynomials it may happen that for only finitely many combinations all polynomial functions take the value zero.

      For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions. If the number of solutions is finite, there are algorithms to compute the solutions. The methods underlying these algorithms are described in the article systems of polynomial equations.

      The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.