Definition:Polynomial over Ring

Let $$\left({R, +, \circ}\right)$$ be a commutative ring with unity whose zero is $$0_R$$ and whose unity is $$1_R$$.

Let $$\left({D, +, \circ}\right)$$ be an integral domain such that $$D$$ is a subring of $$R$$.

Let $$x \in R$$.

Then each element of $$R$$ that can be expressed in the form:

$$a_0 + a_1 \circ x + a_2 \circ x^2 + \ldots + a_n \circ x^n$$

where $$n \in \mathbb{Z}_+$$ and each $$a_i \in D$$, is called a polynomial in $$x$$ over $$D$$.

It is common practice to express a general polynomial in sum notation:

$$\mathbf {Define:} \ \sum_{k=0}^n a_k \circ x^k \ \stackrel {\mathbf {def}} {=\!=} \ a_0 + a_1 \circ x + a_2 \circ x^2 + \ldots + a_n \circ x^n, a_n \ne 0_R$$

... where it is understood that $$a_0, a_1, \ldots, a_n$$ are elements of $$D$$.

Note that it is also understood (as indicated above) that the coefficient of the highest power of the polynomial in $$x$$ is non-zero (although see null polynomial below).

Polynomial Function
If $$R$$ and $$D$$ are both the set of real numbers $$\mathbb{R}$$, 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 $$f: \mathbb{R} \to \mathbb{R}$$ 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.