Binomial Equation

binomial equation

[bī′nō·mē·əl i′kwā·zhən]
(mathematics)
An equation having the form xn-a = 0.

Binomial Equation

 

an equation of the form xna = 0, where a is some real or complex number. The solution of these equations is a consequence of the problem of extracting the nth root of the number Binomial Equation. The binomial equation has n different roots including no more than two real roots. If a is a positive number, then one of these roots—the arithmetic root—is positive. When geometrically representing numbers in the complex plane, all roots of the binomial equation are located on a circle with the center at point O and a radius equal to the arithmetic root of the modulus of the number a (at the apexes of an n-sided polygon).

Binomial equations of the special form xn − 1 = 0 are of great importance. The roots of these equations are called nth roots of unity and have the form

The product and the quotient of two nth roots of unity will also be the nth roots of unity. Among the roots of the nth roots of unity there are such that all the remaining roots are represented as their powers; these roots are called primitive roots. For the root k to be primitive, it is necessary and sufficient that the numbers k and n be relatively prime numbers, namely, that their greatest common denominator be equal to unity. For example, the root 1, is always primitive: k = 1k.

The theory of binomial equations has made it possible to find the conditions for the solubility of the ancient problem of the division of a circle into equal parts using a compass and a straightedge.

REFERENCES

Okunev, L. la. Vysshaia algebra, 2nd ed. Moscow, 1966.
Kurosh, A. G. Kurs vysshei algebry, 9th ed. Moscow, 1968.
References in periodicals archive ?
4) be a binomial equation of y, it is necessary that its coefficients satisfy conditions
When an area similar to the size of Playa Blanca 5 is used in the regression formulae an estimate of almost 34 persons is obtained with the linear model and 43 with the binomial equation.
Oh Gee thanks Nathan and - er - you know if you're training like crazy for one of your big Championship fights and you've forgotten to put in one of your assignments, say, on how Riemannian Binomial equations explain the whole Time-Space Continuum and you haven't got time to swot it all up, just let me know.
In his text he covers algebraic numbers, field extensions, minimal polynomials, multiply generated fields, and the Galois correspondence, closing with such classical topics as binomial equations and solvability in radicals.