Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. They can be represented in the form of f(x) = a_n x^n + a_{n-1 x^n-1 + ... + a_1 x + a_0}, where a_n, a_{n-1, ..., a_0} are constants and n is a non-negative integer.
These functions are essential in various fields, including physics and engineering, as they can model a wide range of real-world phenomena. The degree of a polynomial, determined by the highest power of the variable, influences its shape and behavior, making them a fundamental concept in calculus and graphing.