Cambridge International AS and A Level Mathematics Pure Mathematics 1

Drawing curves

P1^

2

In this example the polynomial x^3 − 3 x^2 − x + 3 has three factors, (x + 1), (x − 1)
and (x − 3). Each of these corresponds to an intersection with the x axis, and to a
root of the equation x^3 − 3 x^2 − x + 3 = 0. Clearly a cubic polynomial cannot have
more than three factors of this type, since the highest power of x is 3. A cubic
polynomial may, however, cross the x axis fewer than three times, as in the case
of f(x) = x^3 − x^2 − 4 x + 6 (see figure 2.30).

Note

This illustrates an important result. If f(x) is a polynomial of degree n, the curve with
equation y = f(x) crosses the x axis at most n times, and the equation f(x) = 0 has at
most n roots.

An important case occurs when the polynomial function has one or more
repeated factors, as in figure 2.31. In such cases the curves touch the x axis at
points corresponding to the repeated roots.

x

f(x) f(x) = x (^3) – x (^2) – 4 x + 6
O
Figure 2.30
x
f(x)
O 4
f(x) = x^2 (x – 4)^2
x
f(x)
O 1 3
f(x) = (x – 1)(x – 3)^2
Figure 2.31