Polynomial function

Summary

A real polynomial function is a function $\R \to \R$ given by a formula $a_0 + a_1 x + \dots + a_n x^n$ with real coefficients. Polynomial functions are continuous and infinitely differentiable, and their derivatives and antiderivatives are again polynomial functions. They are the simplest functions beyond linear ones, and by the Weierstrass approximation theorem they approximate every continuous function on a closed bounded interval uniformly. The definition gives the precise form, and the algebraic side is the polynomial ring.