Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders.

History

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, such as f(x) = x² + 1, 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 a root: this is the statement of the fundamental theorem of algebra.

Related Topics:
Real number - Complex number - Fundamental theorem of algebra

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There is a difference between approximating roots and finding concrete closed formulas for them. Formulas for the roots of polynomials of degree up to 4 have been known since the 16th century (see quadratic equation, Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers for a long time. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). This result marked the start of Galois theory which engages in a detailed study of relations among roots of polynomials.

Related Topics:
Degree - 16th century - Quadratic equation - Gerolamo Cardano - Niccolo Fontana Tartaglia - Niels Henrik Abel - Abel-Ruffini theorem - Galois theory

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The difference engine of Charles Babbage was designed to create large tables of values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's difference method.

Related Topics:
Difference engine - Charles Babbage - Logarithm - Trigonometric function - Newton's difference method

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