Here we discuss polynomials and the more complicated functions that can be defined in terms of them.

### 2.1.1 Polynomials

The *polynomial* function has the general form

(2.1)

where *a*_{i}(*i* = 0, 1, 2, …, *n*) are constants, *n* is a non-negative integer (i.e. including zero), and the symbol Σ means that a sum is to be taken of all terms labelled by the indices 0, 1, 2, …, *n*. The value of *n* defines the *order* (or *degree*) of the polynomial. The expression *x*^{3} − 3*x*^{2} − 6*x* + 8 plotted in Figure 1.1 is therefore a polynomial of order 3.

The *roots* of polynomials are defined as the solutions of the equation

(2.2)

and correspond to the points where a graph of *P*_{n}(*x*) crosses the *x*-axis. For first-order polynomials, (2.2) is a linear equation of the form

(2.3)

where *a* and *b* are constants. This has one root, which is trivially given by *x* = −*b*/*a*.

For second-order polynomials, (