Polynomial

Mathematical definition

Let $P(x)$ be a mathematical function $P:\mathbb{F} \to \mathbb{F}$

If $P(x)$, through algebraic manipulation, can be written in the form of

$$P(x)=\sum_{k=0}^{n} a_k x^k, n\in\mathbb{N},a_0, a_1, \dots, a_n\in \mathbb{F}$$

where $x\in \mathbb{F}$ is an independent variable

Then $P(x)$ is a polynomial over $\mathbb{F}$

Technically, $0$ is a polynomial.

$x+x^{\frac{1}{2}}$ is not a polynomial as $\frac{1}{2}\notin \mathbb{N}$

Degree

The degree of a nonzero polynomial $P(x) \, \deg P=\max\{k: a_k\ne 0\}$.