The discussion of functions of one real variable is continued in this chapter by considering three classes of functions that play an important role in physical sciences. They are: simple algebraic functions; trigonometric functions; logarithms and exponentials.
Here we discuss polynomials and the more complicated functions that can be defined in terms of them.
The polynomial function has the general form
where ai(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 x3 − 3x2 − 6x + 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
and correspond to the points where a graph of Pn(x) crosses the x-axis. For first-order polynomials, (2.2) is a linear equation of the form
where a and b are constants. This has one root, which is trivially given by x = −b/a.
For second-order polynomials, (