📂Functions

Polynomial Function

Definition ¹

A Polynomial of degree $n$ is defined for $n \in \mathbb{N}_{0}$ and $\left\{ a_{k} \right\}_{k=0}^{n} \subset \mathbb{C}$ as follows: $$P(z) := a_{0} + a_{1} z + \cdots a_{n} z^{n} \qquad , a_{n} \ne 0$$

Explanation

A polynomial function is one of the most basic functions that can be considered in all of mathematics, and it has been proven by the Fundamental Theorem of Algebra that there exactly $n$ roots exist.

• According to the definition, a constant function is also a polynomial.
• A polynomial can be differentiated infinitely many times.
• It is a continuous function.

Abstract Algebra

In the notation of abstract algebra, the set of such polynomial functions is denoted as $\mathbb{C}[x]$. Here, the set of coefficients is not necessarily limited to the complex numbers $\mathbb{C}$, and if a field $F$ is given, it can be represented as $F [ x ]$.

The degree of a polynomial can be infinitely large without any issues. In the case of $n = \infty$, the set of such polynomials is denoted as $F[[x]]$.