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Cubic Polynomials with Real Or Complex Coefficients: the Full

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Cubic Polynomials with Real Or Complex Coefficients: the Full Cubic polynomials with real or complex coefficients: The full picture Nicholas S. Bardell RMIT University [email protected] Introduction he cubic polynomial with real coefficients y = ax3 + bx2 + cx + d in which Ta | 0, has a rich and interesting history primarily associated with the endeavours of great mathematicians like del Ferro, Tartaglia, Cardano or Vieta who sought a solution for the roots (Katz, 1998; see Chapter 12.3: The Solution of the Cubic Equation). Suffice it to say that since the times of renaissance mathematics in Italy various techniques have been developed which yield the three roots of a general cubic equation. A ‘cubic formula’ does exist—much like the one for finding the two roots of a quadratic equation—but in the case of the cubic equation the formula is not easily memorised and the solution steps can get quite involved (Abramowitz & Stegun, 1970; see Chapter 3: Elementary Analytical Methods, 3.8.2 Solution of Cubic Equations). Hence it is not surprising that with the advent of the digital computer, numerical root- finding algorithms such as those attributed to Newton-Raphson, Halley, and Householder have become the solution of choice (Weisstein, n.d.). Students studying a mathematics specialism as a part of the Victorian VCE (Mathematical Methods, 2010), HSC in NSW (Mathematics Extension in NSW, 1997) or Queensland QCE (Mathematics C, 2009) are expected to be able Australian Senior Mathematics Journal vol. 30 no. 2 to recognise, plot, factorise, and even solve cubic polynomials as described by ACARA (ACARA, n.d., Unit 1, Topic 1: Functions and Graphs). The latter requirement even goes so far as to encourage students to “solve cubic equations using technology”. Since the application of such technology, e.g., MathCAD (2007), is fundamental to the contents of this paper, it is hoped that the work presented here will prove interesting, relevant, and stimulating to students and teachers alike. It is not the intention here to present a new method for finding the roots of a cubic polynomial; rather, the aim is to allow all the roots to be visualised, their arrangement better understood, and an explanation provided for the complex conjugate root pairing. This understanding becomes of paramount importance when the coefficients are generally complex and any combination 5 of roots can result; an explanation for this latter case appears to be completely Bardell absent from the literature. Cubic polynomials with real coefficients To begin, recall a quadratic function y = ax2 + bx + c can always be cast in vertex form y = a(x − h)2 + k for some constants h and k which means the graph of y = a(x − h)2 + k can be obtained from the standard parabola y = x2 by reflections, dilations and translations. However, this is not so for cubic polynomials since not every cubic graph can be obtained from the standard cubic y = x3 by a combination of reflections, dilations and translations. This failure/inability to express a cubic polynomial in vertex form tends to complicate matters considerably and will be shown to have quite an impact on the following investigation. Consider the general cubic equation y = ax3 + bx2 + cx + d (1) in which the coefficients a, b, c and d are real and a | 0. For completeness, a brief review of cubic polynomials is now given; these salient facts can be found in any good textbook and are presented here without proof. Figure 1 illustrates the three most general forms a cubic function may assume. The increased complexity compared with a quadratic function should immediately be evident. (a) y = x3 – x (b) y = x3 (c) y = x3 + x Figure 1. Illustration of the variation that is possible in cubic curves (PoI = Point of Inflection). In order to produce an accurate graph of a given cubic function, it is necessary to identify four critical points of interest: 1. the location of the relative extremes; 2. the location at which the sign of the curvature (concavity) changes (i.e., the point of inflection (PoI)); 3. the location of the point(s) where the graph crosses the x-axis (i.e., the roots); and 4. the location of the point where the graph crosses the y-axis (i.e., the y-intercept). Australian Senior Mathematics Journal vol. 30 no. 2 6 Cubic polynomials with real or complex coefficients: The full picture The location of the turning points The turning points are found by differentiating Equation (1) and setting the resulting equation for the slope equal to zero, that is, dy = 3ax 2 +2bx +c = 0 (2) dx The location of the turning points is found by using the quadratic formula to solve Equation (2): 2 x = b ± b 3ac (3) 3a 1. If b2 – 3ac > 0, two distinct turning points result—characterised by a local maximum and a local minimum—giving the classic S-shaped cubic curve. For an example, see Figure 1(a). Note that, unlike quadratic curves, the turning points of a cubic are not symmetrically located between x-axis intercepts. 2. If b2 – 3ac = 0, there are no distinct turning points; they have effectively coalesced into a single stationary point of inflection. For an example, see Figure 1(b). 3. If b2 – 3ac < 0, there are no distinct turning points and the function is monotonic. For an example, see Figure 1(c). Further clarification about the nature of the turning points can be found by checking the value of the slope either side of the turning point. Alternatively, the second derivative of Equation (1) may be used: d 2y < 0 the turning point is a maximum If 2 = 6ax +2b dx > 0 the turning point is a minimum The location of the point of inflection Every cubic polynomial has one point of inflection (PoI), that is, a point on the curve at which the sign of the curvature (i.e., the concavity) changes. The PoI is found by setting the second derivative of Equation (1) equal to zero thus: 2 d y b Australian Senior Mathematics Journal vol. 30 no. 2 2 = 6ax +2b = 0, from whichx = (4) dx 3a If, in addition to d 2y 2 =0 dx the third derivative of Equation (1) is non-zero, the existence of a point of inflection is confirmed. This is always the case for a cubic polynomial because d 3y 3 =6a dx and 6a | 0 since by definition a | 0. Hence b x = 3a always defines the location of the PoI on a cubic polynomial. 7 It is further noted that: Bardell t A PoI may be a stationary point, like the one shown in Figure 1(b), but it is not a local maximum or a local minimum. t A PoI is not necessarily parallel to the x-axis. An example of this is shown in Figure 1(c). t If distinct turning points are present, like the ones shown in Figure 1(a), the PoI also corresponds to the mid-point between them. This is evident from Equation (3), which confirms the tuning points are located equal distances either side of x = –b/3a, which is the location of the PoI. The location of the roots A cubic equation has three roots which are found by setting y = 0 in Equation (1) and solving for x. Since the leading coefficient is not zero, then 3 2 if a > 0 lim ax +bx +cx +d = x + if a < 0 3 2 + if a > 0 lim ax +bx +cx +d = x+ if a < 0 Since a polynomial is continuous, then by the Intermediate Value Theorem, if it takes a positive value and a negative value, it must take every value in between, in particular zero. Hence the graph of a cubic must cross the x-axis at least once, and thus one of the roots is always real. The arrangement of the roots can occur in any of the following combinations: t one real root and a pair of complex conjugate roots. t three real roots: – all of which may be coincident; – one of which may be distinct and the remaining pair coincident; – all of which may be distinct. The location of the y-intercept The y-intercept is found by setting x = 0 in Equation (1). This immediately gives yintercept = d. Cubic polynomials in the Cartesian plane To illustrate many of the features discussed above for cubic polynomials with real coefficients plotted in the Cartesian plane, consider the curve y = x3 + 3x2 + x – 5 shown in Figure 2 over the range –4 f x f 2. There is one real root apparent at x = 1 but no indication whatsoever of the whereabouts of the complex conjugate roots. In this particular instance it is easy enough to divide the expression for y by the known factor (x – 1) to yield y = (x – 1)(x2 + 4x + 5) from which the remaining two roots are found to be –2 t i. There are two distinct turning points present which, from Equations (3) and (1), are found to occur at (x, y) = (0.184, –5.089) MIN and (–1.816, –2.911) MAX respectively. Using Equations (4) and (1), the PoI is seen to occur at Australian Senior Mathematics Journal vol. 30 no. 2 8 Cubic polynomials with real or complex coefficients: The full picture (x, y) = (–1, –4), midway between the turning points. The y-intercept is found at y = –5. Figure 2. Plot of the curve y = x3 + 3x2 + x – 5 over the range 4 f x f 2. The complex conjugate roots do not correspond to the locations of either turning point or the PoI shown in Figure 1 and have often been misplaced by well-meaning but ingenuous authors like Stroud (1986, see Programme 2, Theory of Equations, where he [incorrectly] places the complex conjugate roots in the Cartesian plane at one of the turning points of a cubic equation).
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