Definition:Polynomial over Ring

Polynomial Form
Let $$I_X=\left\{1,X,X^2,\ldots\right\}$$ be  the  free monoid on a  singleton $$\{X\}$$.

Let $$\left({R, +,  \circ}\right)$$ be a  ring with unity with  additive identity  $$0_R$$ and multiplicative identity $$1_R$$.

A polynomial form in one variable, or just polynomial over  $$R$$ is a mapping $$f:I_X\to  R:X^n\mapsto  a_n$$ such that $$a_n=0_R$$  for all but finitely many $$n\geq 0$$.

Therefore a polynomial is an ordered triple $$(R,I_X,f)$$. We describe the polynomial as in the indeterminate $$X$$. Often, this singleton is unimportant, and we speak simply of the polynomial $$f$$ over the ring  $$R$$.

Degree
The degree of $$(R,I_X,f)$$ is the supremum


 * $\deg(f)=\sup\left\{n\in\N:f(X^n)\neq 0\right\}$.

Coefficients
Let $$d$$ be the degree of $$(R,I_X,f)$$.

Then the elements of the set $$\left\{a_i:i\leq d\right\}$$ are known as its coefficients.

Leading Coefficient
Let $$d$$ be the degree of $$(R,I_X,f)$$.

The ring element $$a_d$$ is called the leading coefficient of $$(R,I_X,f)$$.

Notation
It follows from Unique Representation in Polynomial Forms that if we let $$a_iX^i$$ denote the polynomial that has value $$a_i$$ on $$X^i$$ and $$0_R$$ otherwise, then $$f$$ can be (uniquely) written


 * $f=a_0+a_1X+\cdots+a_{n-1}X^{n-1}+a_nX^n$.

This is the notation most frequently used when working with polynomials. It is also sometimes helpful to include the tailing zero terms in this sum, in which case


 * $f=\sum_{i=0}^\infty a_iX^i$.

Monic Polynomial
If the leading coefficient $$a_n$$ of $$f$$ is $$1_R$$, then  $$(R,I_X,f)$$ is monic.

Null Polynomial
If $$a_i=0_R$$ for all $$i\geq 0$$, $$(R,I_X,f)$$ is known as the null polynomial or trivial polynomial over $$R$$ in the indeterminate $$X$$.

Mononomial
If $$a_i\neq 0_R$$ for at most one $$i\geq 0$$, then $$(R,I_X,f)$$ is called a mononomial.

Polynomial Equation
A polynomial equation is an equation in the form:
 * $$P_1 \left({x_1}\right) + P_2 \left({x_2}\right) + \cdots + P_n \left({x_n}\right) = 0$$

where each of $$P_1, P_2, \ldots, P_n$$ are polynomials.

Polynomial Function
If $$R$$ and $$D$$ are both the set of real numbers $$\R$$, then the concept of a polynomial function is established.

For a given set of coefficients $$\left\{{a_0, a_1, \ldots, a_n}\right\}$$, the real function $$f: \R \to \R$$ is defined as:


 * $$f \left({x}\right) = \sum_{k=0}^n a_k x^k$$.

The fact that $$f$$ is a function follows from the fact that the Real Numbers form a Field and the operations of addition and multiplication are therefore well-defined.

Also see

 * Definition of Polynomial from Polynomial Ring