IDA: Interactive Document on Algebra

Section 3.1
The notion of a polynomial

Let R be one of the sets Z, Q, R, C or Z/nZ.

Definition

A polynomial over R in the indeterminate X is an expression of the form
a₀ + a₁X + ··· + a_{n - 1}X^{n - 1} + a_nXⁿ,
where n N, a₀, ..., a_n R and X is an indeterminate.
The polynomial is built up from terms a_kX^k for k N.
The elements a₀, ..., a_n are called the coefficients of the polynomial.
The powers X^k of X, for which the coefficient a_k is nonzero, are called the monomials of the polynomial.
We write R[X] for the set of polynomials over R in the indeterminate X.

We also say polynomial in X or polynomial if no confusion is possible.

Two polynomials over R in the indeterminate X are equal if the corresponding coefficients are equal. Polynomials of the form a with a R are called constant.

Using the summation notation we also write

a₀ + a₁X + ··· + a_mX^m =

_k=0^m a_kX^k.

A polynomial in X is often denoted by a symbol like a, but sometimes also by a(X) to emphasize the dependence on the indeterminate X.