Definition:Polynomial
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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 $P$.