Assignment 3
Roots of cubic polynomials
Consider
the cubic equation 
, where a, b, c and d are real coefficients. This equation has either:
(i)
three distinct real
roots
(ii)
one pair of repeated
roots and a distinct root
(iii)
one real root and a pair
of conjugate complex roots
In
the following analysis, the roots of the cubic polynomial in each of the above
three cases will be explored. More specifically, the curve will be plotted in
the xa, xb, xc and xd planes for all the three cases to determine the
conditions under which the roots exist.
As
an illustrative example consider the cubic equation 
which has three
distinct roots as shown in Figure 3.1.
Figure 3.1
CASE 1 Analysis of equation in the xd plane
Consider
the equation 
The
curve 
is plotted in
Figure 3.2 in the xd plane.
A horizontal line will intersect the curve at three
points between the minimum and maximum points. In other words as long as d is lower than the maximum and greater than the
minimum, the equation will have three real roots. For the present example, the
maximum point is (2.53, 6.88) and
the minimum point is (0.13
, -0.06). Thus, as long as 
, the cubic equation will have real roots.
Click HERE for animation
Note
that the value of d does not
influence the turning points in 
. This can observed by considering the stationary points
which are independent of d. The
stationary points occur when 
or at 
.
At the maximum (d = 6.88) and minimum points (d = -0.06) we have repeated roots as shown in Figure 3.3
Figure 3.3
An
alternative way to analyze the equation 
is to
write it as 
.
In
the foregoing example, the equation 
can
be written as 
. Figure 3.4
shows the curve 
and 
for different
values of d.
Figure 3.4
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The equation has three real roots when 
.
CASE 2 Analysis of equation in the xc plane
In the xc
plane, we can consider the equation in the form
In the current, example, 
Figure 3.5 shows
this curve in the xc plane.
Figure 3.5
We
observe that a horizontal line c =
k (k constant) will intersect this curve at three points
when k is less than the maximum at
k = 1.35. Thus, the curve 
will have three real roots when 
. Repeated roots occur at 
as shown in
Figure 3.6.
Figure 3.6
Note
that only part of the curve is shown in Figure 3.6 to focus on the value of c for which repeated roots occur.
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HERE to have a full picture
CASE 3 Analysis of equation in the xb plane
The original equation 
can be cast in
the form
in the xb plane. Here, 
.
Figure
3.8 shows that when 
, the cubic equation has three real roots. Figure 3.9 shows
part of the curve 
when b
= 3.85, 3.86, and 3.87. Observe that
repeated roots occur at b =
3.86.
Figure 3.9
Click HERE for animation
CASE 4 Analysis of equation in the xa plane
In
the xa plane, we consider the
equation 
. In the present example,
Figure 3.10 shows the plot of the curve in the xa plane.
We
observe that when 
, there are three real roots. Repeated roots occur at a = -1.51 and a = 1.06. The graphs
for 
for values of a in the vicinity of -1.51 and 1.06 are shown in Figure
3.11a and 3.11b. Only part of the curves are shown to emphasize how the values
of ÔaÕ influence the nature of the
roots.
Figure 3.11b
Click
HERE for animation
Ajay Ramful
December 2006