Section 3.3
Polynomial functions

In this section, we connect our formal definition of a polynomial with the more usual notion of a polynomial function. Let R be one of the sets Z, Q, R, C or Z/nZ.

Definitions

By replacing the variable X in the polynomial a(X) = a₀ + ··· + a_mX^m R[X] by an element r of R, we find the element

a(r) = a₀ + a₁r + a₂r² + ··· + a_mr^m R.

In this way we obtain a polynomial function

a : R -> R,     r -> a(r).

The element r is called a zero of a(X) if a(r) = 0.

 >>> Remark Examples Gapplet Exercise

The set of polynomial functions is useful for many applications, especially because they are functions which are easy to represent, to manipulate and to use for approximations of other, more complicated, functions.

By way of example, on the next page, we construct polynomial functions with prescribed behaviour.