Definition:Polynomial over Ring

General Ring
Let $$\left({K, +, \circ}\right)$$ be a ring.

Let $$\left({R, +, \circ}\right)$$ be a ring with unity.

Let $$x \in K$$.

Then the expression:
 * $$a_0 + a_1 \circ x + a_2 \circ x^2 + \ldots + a_n \circ x^n$$

where: is called a polynomial over $$R$$ in (a single indeterminate) $$x$$.
 * $$n \in \Z_+$$
 * each $$a_i \in R$$

Integral Domain
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: is called a polynomial in $$x$$ over $$D$$.
 * $$n \in \Z_+$$
 * each $$a_i \in D$$

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


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

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

Coefficients
For a given polynomial, the elements of the set $$\left\{{a_0, a_1, \ldots, a_n}\right\}$$ are known as its coefficients.

Precise definition
The above definition of polynomials as formal linear combinations of an indeterminate is sufficient to describe the algebraic properties we require of a polynomial. However, this definition is circular: one needs a set of polynomials in order to make linear combinations of the elements. Therefore we require a distinction between an indeterminate and a polynomial, which we may achieve in several ways, of which we describe just one.

Let $$I=\left\{1,X,X^2,\ldots\right\}$$ be the free monoid on $$\{X\}$$.

A polynomial over a ring with unity $$\left({R, +, \circ}\right)$$ in the indeterminate $$X$$ is a mapping $$X^n\mapsto a_n$$ of $$I$$ into $$R$$ such that $$a_n=0$$ for all but finitely many $$n\geq 0$$.

Now for each $$r\in R$$, let $$rX^n$$ denote the function that equals $$r$$ on $$X^n$$ and $$0$$ otherwise. Then each polynomial can be written uniquely as a sum


 * $$r_0+r_1X_\cdots +r_nX^n$$

for some $$n\in\N$$ and $$r_i\in R$$. In this way we have a rigorous construction of an algebraic object with the formal properties of the naive definition.

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