Acta Manilana 60 (2012), pp. 15–18
Printed in the Philippines
ISSN: 0065–1370
On sufficient condition for the existence of imaginary
roots of a cubic polynomial equation
Enrico M. Yambao1 & Ma. Carlota B. Decena1,2
1
Department of Mathematics and Physics, College of Science; 2Research Center for the Natural
and Applied Sciences, University of Santo Tomas, España Boulevard, 1015 Manila, Philippines
The exact condition on the parameters a, b, c, d, E R sufficient to show the existence of
imaginary roots of a cubic polynomial equation ax3 + bx2 + cx + d = 0, a ≠ 0 are presented.
Keywords: cubic polynomial equation, imaginary roots
INTRODUCTION
Finding solution of polynomial equations,
either by analytic or iterative approach, has
been the subject of many investigations [1, 3,
4, 6–8]. Particularly of much attention is the
solution of cubic equations with real
coefficients, which serve as basis of important
applications in engineering and applied
sciences. The Van der Waal’s cubic equation
of state, for example, is the most popular
method among chemical engineers for
estimating vapor-liquid equilibrium ratios of
hydrocarbon mixtures at high pressure. In this
method, the roots of the cubic equation of state
are determinant in the actual vapor-liquid
composition of the mixture. Moreover, for
economists, the solution of cubic equations
plays a vital role in their study of the economic
growth rate which is presented as a cubic
function known as the Von Neumann’s model.
In most of these applications, the analysis is
facilitated by characterizing the roots of these
cubic equations [4, 6]. For most of the real world
applications, resulting cubic functions do not
usually appear to have integer coefficients in
which root-finding is performed analytically. In
such cases, iterative approach in finding the
roots is highly relevant. Such methods are
carried out with an assumption that either real
or imaginary roots of the cubic polynomial
equation exist [1, 2, 5].
In this work, we particularly present a sufficient
condition for the existence of imaginary roots
of a cubic polynomial equation with real
coefficients. The derivation is facilitated by
considering the possible graphs of the
corresponding cubic function.
LOCAL EXTREMA OF A CUBIC FUNCTION
Consider the cubic function
f (t) = at3 + bt2 + ct + d
*To whom correspondence should be addressed
E-mail: [email protected]
where a, b, c, d are real numbers and a > 0. Thus,
Acta Manilana • Volume 60 (2012)
Yambao EM & Decena MCB ⏐ Acta Manilana 60 (2012)
i
○
f (t)
○
f i(t) = 3at3 + 2bt2 + c and f ii (t) = 6at + 2b. Since
a > 0 and f i is a quadratic function, the graph of
f i is a parabola that opens upward and its
absolute minimum value occurs at a number
where f ii (t) = 0, that is, at t = –b/3a. Notice that
the root of f ii (t) = 0 also represents the number
where the graph of f has a point of inflection.
Thus, the minimum value of f i is
–b/3a
t
2
f (–b/3a) = 3a(–b/3a) + 2b(–b/3a) + c
= –b2 – 3ac/3a
and for any real number t,
f i (t) ≥ –(b2 – 3ac/3a).
The local extrema of f, if there are any, must occur
at points where f i (t) = 0. Solving f i (t) = 0 yields
Consequently, (1) has an imaginary root only
when f (–b/3a) ≠ 0. Otherwise, –b/3a is a triple
root of (1).
t = –b ± b2 – 3ac/3a
SETTING IN DETERMINING THE
EXISTENCE OF IMAGINARY ROOTS
If b2 – 3ac > 0, then
The existence of an imaginary root of the
resulting cubic polynomial equation
f (t) = at3 + bt2 + ct + d = 0
(1)
consequently results from considering the
following cases:
Case 1.
b2 – 3ac < 0
If b2 – 3ac < 0, then f i (t) ≥ –(b2 – 3ac/3a) > 0.for
any real number t. In this case, f has no local
extrema and must be a monotone whose graph
is a cubical parabola as illustrated in Fig. 1.
Consequently, (1) has one real root and the other
two are imaginary.
Case 2.
b2 – 3ac ≥ 0
If b2 – 3ac = 0, then both f i (t) and f ii (t) are zero
at t = –b/3a and still, f i (t) ≥ 0 for any real number
t. In this case, f is also a monotone whose graph
is a cubical parabola but with horizontal
inflectional tangent as illustrated in Fig. 2.
16
Figure 1. A cubical parabola
f ii (–b ± b2 – 3ac/3a) =
6a (–b ± b2 – 3ac/3a) + 2b = ± 2 b2 – 3ac
By the Second Derivative Test, f has a local
minimum value at –b + b2 – 3ac/3a and has a
local maximum value at –b – b2 – 3ac/3a. Thus,
the graph of f has exactly two turning points
and the graph may cross the horizontal axis in
three possible ways as shown in Figs. 3–5.
All roots of (1) are real if and only if, the local
extrema of f are opposite in sign or one of them
is zero as shown in Fig. 3 and 4, respectively.
Thus, if all roots of (1) are real,
f (–b + b2 – 3ac/3a) f (–b – b2 – 3ac/3a) ≤ 0.
Consequently, (1) has an imaginary root if and
only if, the local extrema of f have the same sign.
This is possible if either
f (–b + b2 – 3ac/3a) > 0
On sufficient condition for the existence of imaginary roots
f (t)
-b/3a
-b / 3a
Figure 2. A cubical parabola with horizontal
inflectional tangent
t
Figure 3. A cubic function with three real roots
f (t)
f (t)
-b / 3a
-b / 3a
t
Figure 4. A cubic function with three real roots with
one of them zero
or,
f (–b – b – 3ac/3a) < 0
t
Figure 5. A cubic function with imaginary roots
Now,
2
Figure 5 shows the graph of a cubic polynomial
satisfying this condition. This is the same
condition in the case b2 – 3ac = 0 if (1) is to
have an imaginary root.
f (–b ± b2 – 3ac/3a)
= a (–b ± b2 – 3ac/3a) 3
+ b (–b ± b2 – 3ac/3a) 2
+ c (–b ± b2 – 3ac/3a) + d
2
= 27a d + 2b3 – 9abc + 2(b2 – 3ac)3/2 / 27a2
17
Yambao EM & Decena MCB ⏐ Acta Manilana 60 (2012)
Thus, f (–b + b2 – 3ac/3a) > 0 implies that
27a2d + 2b3 – 9abc > 2(b2 – 3ac)3/2. Similarly, f
(–b – b 2 – 3ac/3a) < 0
implies
that
27a2d + 2b3 – 9abc < –2(b2 – 3ac)3/2.
Hence, (1) has an imaginary root if and only if,
27a2d + 2b3 – 9abc > 2(b2 – 3ac)3/2
(2)
It can be noticed that (2) is also satisfied when
b2 – 3ac < 0.
The above result is stated:
Theorem: The cubic polynomial equation with
real coefficients at3 + b2 ct + d = 0, (a > 0) has
an imaginary root if and only if
(27a 2d + 2b 3 + 9abc)2 > 4(b2 – 3ac)3.
The above theorem explicitly states the
sufficient condition that guarantees existence
of imaginary roots of cubic polynomial.
FUTURE OUTLOOK
With the guaranty that non-real roots of a given
cubic polynomial exist, the determination of nonreal roots can proceed with iterative approach.
Alternative iterative approach in the
determination of imaginary roots of cubic
polynomial equations is the subject of
investigation in our next paper.
ACKNOWLEDGMENT
It is a pleasure to thank Mr. Mark Louie Ramos
and our colleagues from the Department of
Mathematics and Physics.
18
REFERENCES
[1] Carstensen C & Petkovic M. On iteration methods
without derivatives for the simultaneous
determination of polynomial zeros. Journal of
Computational and Applied Mathematics 1993;
45:251–66.
[2] Ivanisov AV & Polishchuck VK. A method of
finding the roots of polynomials which converges
for initial approximation. Computational
Mathematics on Mathematical Physics 1985;
25(3):1–7.
[3] McLaughlin JB. Covergence of a relaxed
Newton’s method for cubic equations.
Computers & Chemical Engineering 1993;
17(10):971–83.
[4] Petkovic LD & Petkovic MS. Construction of zerofinding methods for Weierstrass functions.
Applied Mathematics and Computation 2007;
184:351–9.
[5] Petkoviæ MS. The self-validated method for
polynomial zeros of high efficiency. Journal of
Computational and Applied Mathematics 2009;
233(4):1175–86.
[6] Petkovic MS. A highly efficient root-solver of very
fast convergence. Applied Mathematics and
Computation 2008; 205:298–302.
[7] Petkovic MS. Some interval iterations for finding
a zero of a polynomial with error bounds.
Computers and Mathematics with Applications
1987; 14(6):479–95.
[8] Petkovic MS, Petkovic L, & Rancic LZ. Point
estimation of simultaneous methods for solving
polynomial equations: A survey (II). Journal of
Computational and Applied Mathematics 2007;
205:32–52.
[9] Petkovic MS & Rancic L. A new fourth-order
family of simultaneous methods for finding
polynomial zeros. Applied Mathematics and
Computations 2005; 164:227–48.