---
tag: 1G0B73WPX8
kind: concept
name: Polynomial function
msc: [26C05]
specializes: [0X3WGX95FE]
sources:
  - url: https://en.wikipedia.org/wiki/Polynomial
---

## 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|0C5TP6RYSV]] 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|0JQE130DHW]] they approximate every continuous function on a closed bounded interval uniformly.
The [[definition|MAM9H3SY3Y]] gives the precise form, and the algebraic side is the [[polynomial ring|0SDC934R52]].

## Explanations

### L1 human

A polynomial function is built from $x$ and constants using only addition and multiplication.
Its graph is smooth with no corners or jumps.
A polynomial function of degree $n \ge 1$ crosses each horizontal line at most $n$ times.
For example $f(x) = x^3 - 2x + 1$ has degree $3$ and $f(1) = 0$.