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 and  additive identity  $$0_R$$.

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 $$a_n=0$$  for all but finitely many $$n\geq 0_R$$.

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  $$R$$.

Coefficients
For a given polynomial, let $$d$$ be the supremum of the non-negative integers such that $$f(X^n)\neq 0$$.

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

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.

Leading Coefficient
Let $$f = \sum_{k=0}^n a_k \circ x^k$$ be a polynomial in $$x$$ over $$D$$.

The coefficient $$a_n \ne 0_R$$ is called the leading coefficient of $$f$$.

Monic
Let $$f = \sum_{k=0}^n a_k \circ x^k$$ be a polynomial in $$x$$ over $$D$$.

If the leading coefficient $$a_n$$ of $$f$$ is $$1_R$$, then $$f$$ is monic.

Null Polynomial
The element $$0_R$$ can be considered to be a polynomial, one such that all $$a_k = 0_R$$.

Such a polynomial is known as the null polynomial or trivial polynomial.

Also see

 * Definition of Polynomial from Polynomial Ring