Integer and polynomial functions

A polynomial function is a function $f: \mathbb{Z} \to \mathbb{Z}$ (or $f: \mathbb{R} \to \mathbb{R}$) defined by $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where the coefficients $a_i$ are integers and $n \geq 0$. When the input is an integer, the output is always an integer.

Important properties

• The sum, difference, and product of two polynomial functions is also a polynomial function.

• Polynomial functions are defined for all integers.

• The degree of a polynomial is the highest power of $x$ with a nonzero coefficient.

• If all coefficients are integers, plugging in integer values for $x$ always gives integer outputs.