# Definition:Polynomial

## Definition

### Real Numbers

A polynomial (in $\R$) is an expression of the form:

$\ds \map P x = \sum_{j \mathop = 0}^n \paren {a_j x^j} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n - 1} x^{n - 1} + a_n x^n$

where:

$x \in \R$
$a_0, \ldots a_n \in \mathbb k$ where $\mathbb k$ is one of the standard number sets $\Z, \Q, \R$.

### Complex Numbers

A polynomial (in $\C$) is an expression of the form:

$\ds \map P z = \sum_{j \mathop = 0}^n \paren {a_j z^j} = a_0 + a_1 z + a_2 z^2 + \cdots + a_{n - 1} z^{n - 1} + a_n z^n$

where:

$z \in \C$
$a_0, \ldots a_n \in \mathbb k$ where $\mathbb k$ is one of the standard number sets $\Z, \Q, \R, \C$.

## Arbitrary Ring

Let $R$ be a commutative ring with unity.

### One Variable

A polynomial over $R$ in one variable is an element of a polynomial ring in one variable over $R$.

Thus:

Let $P \in R \left[{X}\right]$ be a polynomial

is a short way of saying:

Let $R \left[{X}\right]$ be a polynomial ring in one variable over $R$, call its variable $X$, and let $P$ be an element of this ring.

### Multiple Variables

Let $I$ be a set.

A polynomial over $R$ in $I$ variables is an element of a polynomial ring in $I$ variables over $R$.

Thus:

Let $P \in R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial

is a short way of saying:

Let $R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial ring in $I$ variables over $R$, call its family of variables $\left\langle{X_i}\right\rangle_{i \mathop \in I}$, and let $P$ be an element of this ring.

## Term

Let $P = a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0$ be a polynomial.

Each of the expressions $a_i x^i$, for $0 \le i \le n$, is referred to as a term of $P$.