Copyright. Meagan Brooke Vickers

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1 Copyright by Meagan Brooke Vickers 2010

2 The Report Committee for Meagan Brooke Vickers Certifies that this is the approved version of the following report: Methods of Discovering Polynomial Solutions APPROVED BY SUPERVISING COMMITTEE: Supervisor: Co - Supervisor: Efraim Armendariz Mark Daniels

3 Methods of Discovering Polynomial Solutions by Meagan Brooke Vickers, B.S. Math Report Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Arts The University of Texas at Austin August 2010

4 Abstract Methods of Discovering Polynomial Solutions Meagan Brooke Vickers, M.A. The University of Texas at Austin, 2010 Supervisor: Efraim Armendariz Currently, there exist several methods for finding roots of polynomial functions. From elementary processes such as the quadratic formula and the Rational Root Theorem to calculus-based ideas, choosing an appropriate means of solving often depends on the conditions of the given polynomial. This report will explore several solving methods and discuss their advantages as well as their limitations. iv

5 Table of Contents List of Tables... vi List of Figures... vii Chapter 1: Introduction...1 Chapter 2: Current Methods...3 Finding Rational Roots...3 Finding Irrational Roots...4 Chapter 3: Nontraditional Methods...7 Geometry...7 Fixed Point Iteration...10 Chapter 4: New Technologies Expand Knowledge...16 Chapter 5: Evaluating the Reasonableness of Solutions...22 Chapter 6: Conclusion...25 References...26 Vita...27 v

6 List of Tables Table 1: Identifying Roots using the Bisection Method...11 Table 2: Values given by iteration method...12 vi

7 List of Figures Figure 1: Contour Plot of P( x + iy)...4 Figure 2: Euclidean Construction of Roots...7 Figure 3: Carlyle's Method...9 Figure 4: Convergence Seen Through Shrinking Intervals...13 Figure 5: Divergence Seen Through Lengthening Intervals...14 Figure 6: Graph of Figure 7: Graph of Figure 8: Graph of Figure 6 and Figure Figure 9: Graph of Figure 10: Graph of vii

8 Chapter 1: Introduction Solutions of polynomial equations are an important topic covered throughout secondary education and continue throughout students collegiate careers. There are many reliable methods of finding roots learned at the secondary level that are algorithmic in nature and serve to solve polynomials that meet specific criteria, such as polynomials containing only real coefficients. While these methods form a foundation for finding roots of polynomials, they can be expanded upon to create a greater depth of understanding. By revisiting material covered in previous courses, one can use what is already known as a guide to solving more difficult problems. Through nontraditional methods that restrict graphing calculator and computer software use, students can develop an understanding of polynomial solutions and work with mathematics in a whole new way. The development of new technologies has helped to introduce innovative methods of discovering solutions of polynomials with complex coefficients and roots. As stated above, secondary teachers restrict their curriculum to polynomials with real coefficients, but college professors will expect students to have an understanding of complex coefficients. If teachers are able to mix both conventional and unconventional methods of finding both real and complex roots, students may enter college with knowledge of several strategies that will work across multiple problem scenarios and possess problem solving skills that can be used across various contents. This paper will address and expand upon current solution methods for finding polynomial roots and explore options that might enhance student understanding. From calculator to noncalculator methods, real and complex coefficients, and real and complex roots, the idea is 1

9 to give secondary teachers an opportunity to explore other options that can be worked into current curriculum to promote students knowledge and their college readiness. 2

10 Chapter 2: Current methods Finding Rational Roots The Rational Root Theorem is commonly used to find all rational roots of a polynomial equation with integral coefficients. The theorem states: Given a polynomial of the form, where and, 0, the roots will be of the form where and. One can then list all of the possible rational roots by dividing all factors of by all factors of. Depending on the values of both the leading coefficient and the constant term, there may be occasions where more than ten possible roots will require testing using synthetic division. Descartes Rule of Signs is a process that may return the result of certain possible roots being eliminated, but is not always helpful and can create more work. Several other theorems, such as the quadratic formula, allow for both real and complex roots to be identified but these methods can only be used under specific criteria; e.g. polynomial coefficients are real, all complex roots are given, or the polynomial only has two complex roots. While these methods provide solvers with a starting point, the complexity of many polynomial solution problems do not fit into the category of being solvable by such elementary processes. Combs and Walls offer a suggested method that can significantly reduce the amount of time necessary to solve for roots of polynomials. Combs and Walls propose that when using synthetic division to check for rational roots, a rational number can be eliminated as a possible root if a noninteger appears in the quotient [3, p. 16]. The proof of this suggestion is given in [3, p. 17]. 3

11 Finding Irrational Roots Combs and Walls method is used specifically for finding rational roots, therefore one must look to other methods in order to find non-rational roots. Barrs and the Braseltons have another suggested method of finding roots of higher degree polynomials [1]. First, one employs the Fundamental Theorem of Algebra, which states that every polynomial of degree n with complex coefficients has n complex zeros (including multiplicities). The mathematics can be challenging if the zeros are all complex, such as in Figure 1. The zeros for this sixth-degree polynomial correspond to the level curves that are points [1, p.380]. Figure 1. A contour plot of,, 5 5 [1, p. 380]. Barrs and the Braseltons extend the Rational Root Theorem and show that for polynomials with integral coefficients the factors of the constant term and leading coefficient can be written as a sum of squares and can often lead to finding complex zeros [1, p. 381]. The Imaginary Rational Root Theorem states that given the polynomial function with real or complex coefficients, such as in (1.1), rational complex numbers 4

12 such that 0 are zeros because when substituted yield a remainder of zero. This verification could also be completed using synthetic substitution. The rational complex zeros are of the form, for integers p, q, r. For a polynomial with integral coefficients, it follows that and subsequently 2. It is known that the resulting polynomial is a factor of P(x) [1, p. 381]. Because it is a factor, P(x) can be written as 2, where is an integer for each k. It can be seen that and 5

13 . Since is a divisor of and is a divisor of, it has been shown that the Imaginary Rational Root Theorem holds for this situation [1, p. 381]. 6

14 Chapter 3: Nontraditional Methods Geometry One way to expand on current methods for finding polynomial roots is through the use of geometry. Some scholars restrict the use of technologies because they want to focus on the reasons behind the mathematics. Hornsby notes that as technology advances, students are becoming less responsible for calculations through time consuming methods and are relying more on computers and calculators [6, p. 362]. According to Hornsby, Euclid s Elements includes the drawing in Figure 2 which assists readers in their understanding of how to find roots using only simple geometry and algebra [6]. Figure 2. Euclidean construction of roots [6, p. 363]. Hornsby explains how the roots are found, starting with properties of similar right triangles. Some mathematicians have dubbed this topic HLLS and SAAS. The author will focus on SAAS; that is, for two right triangles that share a common side, the ratio of the segment of one triangle to the altitude is equal to the ratio of the same altitude to the 7

15 segment of the second triangle. From this use of high school geometry it can be stated that and it follows that c CB,. Since, then. It follows that 0. A similar method affords the opportunity to incorporate prior knowledge across different curricula, specifically circles. Hornsby describes Carlyle s Method, one in which the equation 0 can be solved by finding the points of intersection of the x-axis with a circle having a diameter with endpoints at (0, 1) and (-b, c) as shown in Figure 3 [6, p. 363]. 8

16 Figure 3. Carlyle s Method [6, p. 364]. If there are two solutions to the quadratic, then the circle will intersect the x-axis at two different points, and the x-values of these points are the solutions to the quadratic. If there is a single solution, the circle will bounce off of the x-axis creating a point of tangency and thus a double root. If the quadratic equation has no real roots, then the circle will not intersect the x-axis [6, p.364]. This method can be explained by algebra. The equation of the circle in standard form is given by , 4 which simplifies to:

17 Now setting y = 0, it is clear that the x-values of the intersections of the circle are of the form x 2 bx c 0. According to Schultze, there exist two additional methods for finding roots to the equation 0 through graphing. One option is to graph the line against the fixed graph. The x-values, or abscissas, of the points of intersection of the line and parabola provide the roots to the equation. The second option provided is to graph the rectangular hyperbola 1 and, which is obtained using partial substitution. Again, the abscissas of the points of intersection are the solutions to the quadratic [6, p. 365]. Fixed Point Iteration Some may consider solutions of polynomials to be a topic covered only at the basic Algebra I and Algebra II levels, but secondary educators may find that some alternate routes for evaluating roots are better left to calculus students. One way to begin a calculus course is to find roots of polynomials through fixed point iteration, which 10

18 combines a precalculus review, a discussion of slope in secant and tangent lines, and an investigation of limits [2, p. 2]. If one were given the following polynomial (2.1) and asked to find the roots, a common place to begin would be by identifying integers that the roots lie between. In an attempt to obtain a high degree of accuracy, we will use the bisection method. The bisection method is a root finding algorithm which repeatedly bisects an interval and then selects a sub-interval in which the root must lie. Given that f(0) is positive, f(1) is negative and that (2.1) is continuous, by the Intermediate Value Theorem we know that a root lies on the interval [0, 1]. We will then bisect the interval and see that f(0.5) is negative. Therefore the root must lie on [0, 0.5]. Now, choosing a value on [0, 0.5], we see that f(0.3) is positive, so the root must lie on the interval [0.3, 0.5]. By continuing this process, the following table can be produced. Table 1. Identifying Roots using the Bisection Method [2, p. 2] x While this method will eventually get one close to the root and can be done with only a scientific calculator, it is tedious and can take a long time to find the root to a high degree of accuracy. At this point, one can employ fixed point iteration. Fixed point iteration is a recursive method that allows one to find fixed points of an equation, that is 11

19 where P(x) = x. In order to use this method we will alter the function by solving for the variable x. Now (2.1) can be rewritten as 1 3. (2.2) We are altering the function so that instead of finding the roots in a traditional way, we are finding intersections of an altered function with the line y = x. Therefore, the fixed points of (2.2) will give zeros of (2.1). Knowing that a root exists on [0, 1], start with an initial guess on that interval. Iteration gives the following table. Table 2. Values given by iteration method [2, p. 3] Notice that iteration can cease when two consecutive iterates are identical. Once the input and output values match, each successive iterate will render the same output. This is because we have found an approximate value where (2.2) intersects the line y = x. One must consider where to begin the iterative process, because different seed, or initial, values will render different output values through iteration. Notice that (2.1) has three different solutions yet the iterative process only produced one value. Depending on the initial value chosen, the outputs might either diverge to infinity or converge to a root. There is a way to choose a seed that will converge, but first it is worthwhile to 12

20 graphically examine what is meant by convergence. Fixed point iteration will produce a root when a given interval of output values is shorter than the length of the interval containing the original input values, as shown in Figure 4. Notice that the original interval of input values is length 0.1, while the first iteration gives an interval of length.025. After a second iteration, the length of the interval then shrinks to which is approximately one-seventh of the original interval [7]. Here, we see convergence to a fixed point. Figure 4. Convergence Seen Through Shrinking Intervals [2, p. 4] In Figure 5, observe a seed that diverges. The consecutive output values create longer intervals than previous input intervals, therefore iteration beginning with this seed diverges to infinity. 13

21 Figure 5. Divergence Seen Through Lengthening Intervals [2, p. 4] Given that convergence will occur when 2, we have that 1. 2 After a discussion of limits and the significance of lim 0, one can introduce the first derivative as the magnification factor (MF). The magnification factor is given by 14

22 lim 2, and this definition leads naturally to the theorem stating The fixed point iteration algorithm converges if 1 for values of x near the initial guess of [2, p. 5]. While seed values outside the intervals given by the magnification factor might converge to a root, the values within intervals given by the MF guarantee convergence. One might also choose another way to alter the function, perhaps by , each of which might yield a different initial root than (2.2). Once a root is found, the last two roots can be identified using synthetic division and solving the resulting quadratic using another method [2, p 6]. All of these methods require use of prior knowledge to find solutions of polynomial equations. Mathematicians encourage users to expand what they know to assist in finding what they do not. 15

23 Chapter 4: New Technologies Expand Knowledge While Hornsby is of the opinion that calculator use should be restricted, Grishin strongly encourages the use of emerging, 21 st century technologies as tools to solve more complex polynomial equations. Grishin takes technology far beyond the confines of the graphing calculator. One must note that while calculators are able to find roots in a designated graphing window with a high degree of accuracy, some polynomials are difficult to solve implicitly, specifically functions with nonreal coefficients. However, if the coefficients of the function are real, then all real roots will be seen in some viewing window, while all other roots will come in pairs as complex conjugates of each other [5, p. 19]. Grishin offers graphing techniques that do not have restrictive hypotheses; that is, a polynomial equation can be of any arbitrary degree, have real or complex coefficients, and have both real and complex roots. In order to illustrate this technique we will first explore a problem that can be solved for one of the variables and graphed using a graphing calculator. Grishin requires that all equations are in terms of the variable z, where z is a complex number of the form, such that x and y are both real [5, p. 19]. Coefficients are also of the complex form, thus the polynomial equation 0 is complex throughout. Suppose one were asked to solve the following equation, given the aforementioned stipulations: 16

24 Because, we have equivalency with After expanding the equation, separating the real and imaginary parts, and setting each equal to zero, we have the real equation 0, which factors to give the form 1 0, (5.1) hence resulting in lines and 1. This graph is shown in Figure 6. We also have the imaginary equation which can be solved for y, producing an equation that can be readily displayed using a graphing calculator, (5.2) 17

25 This equation has a zero at x = 5, a vertical asymptote at, and a horizontal asymptote at y = - 1. The two equations are graphed separately in Figures 6 and 7, and together in Figure 8. Figure 6. Graph of 5.1 [5, p. 20] Figure 7. Graph of 5.2 [5, p. 20] 18

26 Figure 8. Graphs of 5.1 and 5.2 [5, p. 21] The roots of the original equation are found at the intersections of these graphs, that is, where the real and imaginary parts cross. Therefore, roots occur at 2, 1 2. Figure 8 is known as the Argand diagram, as it is the plot of the real and imaginary parts of complex numbers on the x- and y-axis, respectively [5, p. 21]. The usefulness of the Argand image is given in several further examples. While the previous example was chosen specifically because it could be solved with only a graphing calculator, complex and higher level problems are ideal to solve using this method. The only requirement is that one has access to a graphing utility that allows equations to be inserted implicitly rather than as a y as a function of x form. After completing the same algebraic steps of substituting, expanding, and isolating real and complex parts, the following equation is a cubic equation with no real roots 19

27 (5.3) Figure 9. Graph of 5.3 [5, p. 21]. The roots are found at the intersections of the real and imaginary relations. That is, 1,,1. For a quartic equation with two real roots and one complex root with multiplicity two, given by , (5.4) the graphs of the real and imaginary parts are created using the graphing software mentioned by Grishin, giving the Argand image in Figure 10, and roots 1,1,. 20

28 Figure 10. Graph of 5.4 [5, p. 21]. These equations could not be solved using previously stated methods due to the nature of the coefficients and the roots, however they can be solved with advanced technological software. 21

29 Chapter 5: Evaluating the Reasonableness of Solutions It is necessary that we spend a moment discussing methods for evaluating the reasonableness of one s answers. Due to the fact that some polynomial solutions are difficult to check by conventional methods of substitution, Gillman provides a method for evaluating the correctness of complex solutions without using -notation [4]. Given 3 5 4, let u be the complicated part of the quadratic formula that results in roots being complex. That is: Note that Now the solutions to the equation given by the quadratic formula can be written as or (4.1) (4.2) 22

30 By verifying whether or not (4.1) is a root, we can substitute and obtain Then verifies that (4.1) is a root. Another simple way of checking for errors is the result of a theorem which states that the sum and product of the roots of a polynomial should be given by and, respectively. One can verify that the two roots (4.1) and (4.2) yield the designated sum and product. That is, for solutions , the sum of the roots is given by , 23

31 and the product is given by It has been verified that (4.1) and (4.2) are roots of the given polynomial. The converse of this theorem is a theorem itself and provides yet another means of verifying solutions. If the sum of the roots, then. 24

32 Chapter 6: Conclusion Systematic processes for finding roots of polynomials have been around for thousands of years. The Babylonians developed the quadratic formula around 2000 B.C. while sixteenth century mathematicians succeeded in finding similar formulas for polynomials with degrees three or four. Ruffini laid the groundwork for the impossibility theorem (later proved by Abel), stating that one cannot derive a formula for polynomials of degree greater than four [7, p. 188]. As a result of this theorem, there exist alternative methods for solving such polynomials. Listed here are just a few methods that can be applied to higher-degree polynomials as well as polynomials with real or complex coefficients. Through the use and expansion of prior knowledge, users can develop an enriched understanding of one of the major topics in mathematics that permeates secondary education. The development of innovative technology allows for new and alternative methods that were previously inaccessible. Here, the central idea behind solving is to first find a process that will work with the conditions of the polynomial, then solve the polynomial in the most efficient way possible, and finally evaluate the reasonableness of one s solutions. While all procedures possess advantages and limitations, each has merit in specific circumstances. 25

33 References 1. Sharon Barrs, James Braselton, Lorraine Braselton, A Rational Root Theorem for Imaginary Roots, The College Mathematics Journal 34, No. 5, (2003) Thomas Butts, Fixed Point Iteration An Interesting Way to Begin a Calculus Course, The Two-Year College Mathematics Journal 12 (1991) Randel Combs and Gary L. Walls, Rational Roots of Polynomials with Integer Coefficients, Mathematics and Computer Education 38, No. 1 (2004) Leonard Gillman, The Roots of a Quadratic, The College Mathematics Journal 33 (2002) Anatole Grishin, Graphical Solution of Polynomial Equations, Australian Senior Mathematics Journal 23 (2009) E John Hornsby, Jr. Geometrical and Graphical Solutions of Quadratic Equations, The College Mathematics Journal 21 (1990) Victor Y. Pan, Solving a Polynomial Equation: Some History and Recent Progress, Siam Review 39 (1997)

34 Vita Meagan Vickers received a Bachelor of Science in Mathematics from the University of Texas at Austin in 2006 and a Master of Arts in Mathematics in She currently resides in Austin, Texas and teaches high school mathematics. address: meagan.vickers@gmail.com This report was typed by Meagan Vickers. 27

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