Contents How You Ma Use This Resource Guide ii 16 Higher Degree Equations 1 Worksheet 16.1: A Graphical Eploration of Polnomials............ 4 Worksheet 16.2: Thinking about Cubic Functions................ 5 Worksheet 16.3: Finding Irrational Roots: An Application of the Locator Theorem 6 Worksheet 16.4: Curve Fitting with Polnomials................. 9 Worksheet 16.5: A Graphical Eploration of Rational Functions........ 11 Worksheet 16.6: The Salt Tank.......................... 12 Answers 13 i
How You Ma Use This Resource Guide This guide is divided into chapters that match the chapters in the third editions of Technical Mathematics and Technical Mathematics with Calculus b John C. Peterson. The guide was originall developed for the second editions of these books b Robert Kimball, Lisa Morgan Hodge, and James A. Martin all of Wake Technical Communit College, Raleigh, North Carolina. It has been modified for the third editions b the author. Each chapter in this Resource Guide contains the objectives for that chapter, some teaching hints, guidelines based on NCTM and AMATYC standards, and activities. The teaching hints are often linked to activities in the Resource Guide, but also include comments concerning the appropriate use of technolog and options regarding pedagogical strategies that ma be implemented. The guidelines provide comments from the Crossroads of the American Mathematical Association of Two-Year Colleges (AMATYC), and the Standards of the National Council of Teachers of Mathematics, as well as other important sources. These guidelines concern both content and pedagog and are meant to help ou consider how ou will present the material to our students. The instructor must consider a multitude of factors in devising classroom strategies for a particular group of students. We all know that students learn better when the are activel involved in the learning process and know where what the are learning is used. We all sa that less lecture is better than more lecture, but each one of us must decide on what works best for us as well as our students. The activities provided in the resource guide are intended to supplement the ecellent problems found in the tet. Some activities can be quickl used in class and some ma be assigned over an etended period to groups of students. Man of the activities built around spreadsheets can be done just as well with programmable graphing calculators; but we think that students should learn to use the spreadsheet as a mathematical tool. There are obstacles to be overcome if we are to embrace this useful technolog for use in our courses, but it is worth the effort to provide meaningful eperiences with spreadsheets to people who probabl will have to use them on the job. Whether or not ou use an of the activities, we hope that this guide provides ou with some thought-provoking discussion that will lead to better teaching and qualit learning. ii
Chapter 16 Higher Degree Equations Objectives After completing this chapter, the student will be able to: Use snthetic division to find roots of polnomials; Reduce a polnomial to linear and quadratic factors using snthetic division to find all the roots of a polnomial; Determine the possible rational roots and test them using snthetic division; Find irrational roots using a numerical method and technolog; model applications with polnomial functions and use the function to solve a problem; Find vertical and horizontal asmptotes of rational functions; Solve equations that involve rational functions. Teaching Hints 1. Before finding zeros of polnomial functions, students should be given some graphical eperiences with higher degree polnomials. To find all the roots of a polnomial, it is a great help to know the possible shapes of the graph of that polnomial. Through graphical eploration with the use of a graphing utilit, have students find all the possible shapes of a third-, fourth-, and fifth-degree polnomial. (Refer to Activit 16.1) 2. Treat snthetic division as an algebraic method of confirming a root of a polnomial and as an algebraic method of factoring a polnomial. Division is used to reduce a polnomial into linear and quadratic factors, which we alread have an algebraic method for solving. 3. Introduce snthetic division with a polnomial function in contet of some application. Consider the eample on the net page. 1
Peterson, Technical Mathematics, 3rd edition 2 Eample 16.1 T (t) = 0.05t 3 1.8t 2 + 14.4t + 14 represents the temperature on a winter da where t = 0 corresponds to 5 A.M. and 0 < t < 24. Use groups to divide the function T (t) b the factors t 2, t 6, t 8, t 12, t 14, t 16, t 20, and t 24. Using a graphics utilit, graph the temperature polnomial and note that the remainder generated b snthetic division is the temperature at the corresponding time. Algebraicall confirm the result. 4. Use a polnomial in contet to find rational and irrational roots. The polnomial given above, which represents temperature, has one rational root and two irrational roots (one which is out of the realistic domain). Use such a polnomial to demonstrate that the rational root theorem onl gives candidates for rational roots and a numerical method is needed to find the irrational roots, ecept when the polnomial can be reduced to quadratic factors b division. 5. Use a spreadsheet or write a program for finding irrational roots (actuall an real root). The bisection method is another valid numerical method for finding roots. Also, the bisection method is intuitivel easier to understand than the linear interpolation method. (Refer to Activit 16.2) 6. Model applications with polnomial functions. The tet includes some applications that require the use of geometr formulas to model the volume of a bo, a silo, and a propane tank (see pages 739 742). However, ever polnomial is NOT derived from a geometr formula. Therefore, model data sets with polnomial functions b solving a sstem of linear equations. It gives students a real sense of mathematical power to be able to arrive at a polnomial model from data. Also, it connects sstems of equations, matrices, and polnomials within the realm of a realistic application. (Refer to Activit 16.3) 7. Through graphical eploration of rational functions and some pattern recognition, allow students to determine on their own an algebraic method of finding vertical and horizontal asmptotes and -intercepts. (Refer to Activit 16.4) 8. Illustrate the need to stud rational functions and to solve equations that involve rational functions with the use of applications. For eample, Eample 16.2 The population densit (in people per square mile) of each of two cities miles from the center of each cit are given as follows: D 1 = 900 2 + 20, D 2 = 2000(2 + ) 40 2 + 20 Determine at which mile from the center of each cit Cit 1 is more dense than Cit 2. Also, as ou travel farther from the center of each cit, how do the two densities change? What do the approach? (Refer to Activit 16.5 for another application)
Peterson, Technical Mathematics, 3rd edition 3 Guidelines The emphasis of this chapter should be on thinking of polnomial and rational epressions as FUNCTIONS. According to the Crossroads document, increased attention should be given to determining the real roots of an equation b a combination of graphical and numerical methods and to learning to distinguish between classes of functions and use functions in modeling situations. It is important for students to know the graphical characteristics of polnomial and rational functions so the can anticipate the number of possible real zeros, the behavior of the function as gets larger, and most importantl, relate a particular polnomial or rational function to a scatter plot of a realistic data set to be modeled. This chapter has tremendous potential to build man connections between topics in mathematics (matrices, sstems of equations, curve fitting, algebraic, graphical, and numerical methods of solving equations to make predictions) and between mathematics and other fields of stud. Increased Attention Pattern recognition,, drawing inferences Use of functions in modeling Interpret the behavior of graphs of functions near asmptotes and for ver large and ver small values of the variable Guidelines for Content Decreased Attention Rote application of formulas Graphing functions with paper and pencil Equation solving strategies such as upper and lower bounds theorem and Descartes rule of signs Activities 1. A Graphical Eploration of Polnomials 2. Finding Irrational Roots: An Application of the Locator Theorem 3. Curve Fitting with Polnomials 4. A Graphical Eploration of Rational Functions 5. The Salt Tank
Peterson, Technical Mathematics, 3rd edition 4 Student Worksheet 16.1 A Graphical Eploration of Polnomials The object of this activit is for ou to make generalizations about the graphical shapes of third-, fourth-, and fifth-degree polnomials. This activit should be done in groups of 3-4 students so each group member can produce a different graph for the group to analze. Using a graphing utilit and changing one parameter at a time, graph each polnomial and record a rough sketch of its graph. Then state conclusions about all the possible shapes of the polnomial. Eercises 1. The Cubic: = a 3 + b 2 + c + d Set 1: = 3 = 3 Set 2: = 3 + 2 = 3 + 4 2 = 3 + 8 2 = 3 2 = 3 4 2 = 3 8 2 Set 3: = 3 + = 3 + 4 = 3 + 8 = 3 + = 3 4 = 3 8 Set 4: = 3 + 2 + = 3 + 4 2 + 4 = 3 + 8 2 + 8 = 3 2 = 3 4 2 4 = 3 8 2 8 Set 5: = 3 + 2 + + 1 = 3 + 2 + + 4 = 3 + 2 + 4 Your conclusions on the third degree polnomial: 2. The Quartic: = a 4 + b 3 + c 2 + d + e Set 1: = 4 = 4 Set 2: = 4 + 3 = 4 + 4 3 = 4 4 3 Set 3: = 4 + 2 = 4 + 4 = 4 4 2 Set 4: = 4 + = 4 + 4 = 4 4 Set 5: = 4 + 4 3 + 4 2 + 4 = 4 4 3 + 4 2 4 = 4 + 4 3 4 2 + 4 = 4 4 3 4 2 + 4 = 4 4 3 4 2 4 = 4 + 4 3 4 2 4 Set 6: = 4 4 3 4 2 4 + 4 = 4 + 4 3 4 2 4 4 Your conclusions on the fourth-degree polnomial: 3. The Fifth-Degree Polnomial Make our own list of fifth-degree functions to analze, then graph them to determine all the possible shapes of fifth-degree polnomials.
Peterson, Technical Mathematics, 3rd edition 5 Student Worksheet 16.2 Thinking about Cubic Functions 1. Draw a rough sketch of the following cubic functions, if possible: (a) A cubic with onl one root, and the real root is negative. (b) A cubic with onl one root, and the real root is positive. (c) A cubic with three real roots, all positive. (d) A cubic with three real roots, one positive and two negative. (e) A cubic with three real roots, and smmetr with respect to the origin. (f) A cubic whose graph has smmetr with respect to the -ais. (g) A cubic with three real roots, but two of them are equal. (h) A cubic with two and onl two different roots. 2. Eplain wh it is impossible for a cubic function to have two and onl two roots. 3. Eplain wh it is impossible for a cubic function to have a graph that has smmetr with respect to the -ais.
Peterson, Technical Mathematics, 3rd edition 6 Student Worksheet 16.3 Finding Irrational Roots: An Application of the Locator Theorem The Locator Theorem If P is a polnomial and if P (a) and P (b) have opposite signs, then there is an -value between a and b such that P () = 0. Appling the Locator Theorem to Spreadsheets Using a spreadsheet ou can find man values of P () between a and b, which enables ou to find two other values closer together sa, c and d so that P (c) and P (d) have opposite signs. Continuing this process, we can sandwich the -intercept that is the solution to the equation P () = 0. a c d b P() FIGURE 16.3.1 Eample 16.3.1 Use a spreadsheet to find the real solution to 3 2 = 3 rounded to three decimal places. Solution Setting the equation equal to 0 results in 3 2 3 = 0. Using a spreadsheet, find values of P () = 3 2 3 between = 5 and = 5 incrementing b 1, as in Figure 16.3.2. Notice that P (1) = 4 is negative and P (2) = 1 is positive; therefore a solution eists between = 1 and = 2. Now, change the increment to 0.1 and find values of P () = 3 2 3 between = 1 and = 2, as in Figure 16.3.3. Notice that P (1.8) = 0.768 and P (1.9) = 0.059 have opposite signs, therefore a solution eists between = 1.8 and = 1.9. FIGURE 16.3.2 FIGURE 16.3.3 Continue this process until three decimal places are the same for two -values that have opposite-signed P () values. You should find the approimate solution = 1.893, as shown in Figure 16.3.4 on the net page.
Peterson, Technical Mathematics, 3rd edition 7 FIGURE 16.3.4 The Bisection Method and Programming This numerical process begins like the previous method b finding a and b such that P (a) and P (b) have opposite signs, like the curve in Figure 16.3.5. Once two such values have been found, ou bisect the interval between a and b, sa = c, and find P (c). If P (c) and P (a) have opposite signs, then a solution eists between a and c. Again ou bisect the interval between a and c, sa = d, and find P (d). If P (a) and P (d) have opposite signs, then a solution eists between a and d. But if P (a) and P (d) have the same sign, then the solution is in the other half of the interval between = c and = d. Continuing this process will allow ou to zero in closer and closer to the -value that makes P () = 0. P() a d c b FIGURE 16.3.5 Eample 16.3.2 Find the real solution to 3 2 3 = 0 using the bisection method. Solution First, find two -values with opposite-signed P () values. P (1) = 4 and P (2) = +1, so ou bisect the interval [1, 2]. You now need to find P (1.5), as shown in Figure 16.3.6. P (1.5) = 2.625 and P (1) = 4, therefore the solution is in the interval [1.5, 2], since P (1) and P (1.5) were both negative. Bisecting the interval [1.5, 2], ou now need to find P (1.75). P (1.75) = 1.141 and P (1.5) = 2.625, therefore the solution is in the interval [1.75, 2]. (See Figure 16.3.7.) FIGURE 16.3.6 FIGURE 16.3.7
Peterson, Technical Mathematics, 3rd edition 8 Bisecting [1.75, 2], ou need to find P (1.875). P (1.875) = 0.1582 and P (1.75) = 1.141, therefore the solution is in the interval [1.875, 2], as in Figure 16.3.8. Continuing this process, ou can find the solution to a reasonable degree of accurac. FIGURE 16.3.8 Eercises 1. Write a program that applies the bisection method or develop a spreadsheet template that can be used to solve equations numericall. Then use the program or spreadsheet to solve each equation or application below. Remember to write the equation as f() = 0 before numericall solving. 2. The bending moment of a beam with a load on one end is given b the polnomial M() = 0.08 4 2.3 3 + 14.8 2 28.5, where is the distance in ards from the attached end of the beam. Find the smallest number of ards from the end where the bending moment is 10. 3. The growth of the AIDS virus from 1982 to 1985 was statisticall found to be N(t) = 69t 3 +298t 2 +1286t+920, where N represents the number of new cases per ear and t is the number of ears after 1982 (1982 corresponds to t = 0). (a) If the growth continues to climb according to this function, in what ear will the number of new AIDS cases be 22,000? (b) In what ear will there be 42,000 new AIDS cases? (c) Do ou believe the number of new AIDS cases will reach 100,000 cases per ear? If so, in what ear? ( ) 355 4. The cost, in cents, of the materials for the construction of a 12-oz soda can is given b C(r) = 0.1r 2 +0.04, r where r is the radius of the can in cm. Determine the radius of the soda can that could be constructed for a materials cost of 9 cents. 5. The volume of two combined sound waves is modeled b the function V (t) = sin(110πt) + sin(120πt). (a) Does the volume ever reach a magnitude of 1.8? (b) If so, then how man times and for which times does it reach 1.8 within the times 0 < t < 0.03?
Peterson, Technical Mathematics, 3rd edition 9 Student Worksheet 16.4 Curve Fitting with Polnomials Eample 16.4.1 Given three points (1, 2), (3, 8), and (6, 4), find a second-degree polnomial function that passes through the three given points. Solution Substituting the three points into the second-degree polnomial P () = a 2 + b + c, the sstem of three equations in a, b, and c can be written. a + b + c = 2 9a + 3b + c = 8 36a + 6b + c = 4 Solving this sstem above ou get a = 0.8667, b = 6.4667, and c = 3.6. Therefore the polnomial that passes through the three given points is = 0.8667 2 + 6.4667 3.6. Eample 16.4.2 Given four points (1, 2), (3, 8), (5, 1), and (6, 4), find a third-degree polnomial function that passes through the four given points. Solution Substituting the four points into the third-degree polnomial P () = a 3 + b 2 + c + d, the sstem of four equations in a, b, c, and d can be written. a + b + c + d = 2 27a + 9b + 3c + d = 8 125a + 25b + 5c + d = 1 216a + 36b + 6c + d = 4 Solving the sstem above ields a = 0.7583, b = 8.45, c = 26.942, and d = 17.25. Therefore the polnomial that passes through the four given points is = 0.7583 3 8.45 2 + 26.942 17.25. Generalization: To find a fourth-degree polnomial, 5 points are needed to generate a sstem of equations in order to solve for the coefficients. How man points are needed for a fifth-degree polnomial? an nth-degree polnomial? Eercises 1. The following data is from a local power compan and was used to estimate the efficienc of a generator. Find three possible polnomial models that could be used to predict the heat rate of a generator,, in BTU/MWH, based on the wattage output of the generator,, in MW. Show the selected points ou use for each polnomial, all three polnomial models generated, and the residuals for each polnomial. Choose the best polnomial of the three that fits the data, eplain wh it is the best, and use it to solve the problem below the data. Wattage output, 90 112 134 155 165 175 185 194 203 212 Heat rate, 9820 9670 9585 9530 9510 9505 9510 9520 9535 9540 Use the polnomial that best fits the data and determine the level at which to run the generator so that the heat rate will be minimized. (You will need to use a graphing utilit to answer this question.)
Peterson, Technical Mathematics, 3rd edition 10 2. Use a welding rod, a weight, and 2 meter sticks, as demonstrated in Figure 16.4.1, to gather several data points that correspond to (1) the horizontal distance from the attached end to the location at which the deflection of the beam will be measured,, and (2) the vertical deflection of the beam,. FIGURE 16.4.1 (a) Find at least three different polnomials using an appropriate selection of the data points measured. Show the points ou use for each polnomial, all three polnomial models, and the residuals for each polnomial. Choose the best polnomial that models the entire data set, and eplain wh it is the best model. (b) Describe the domain and the range for the best polnomial function as it relates to the application. (c) Predict the deflection of the beam at a distance 26 inches from the attached end. Also, predict the location from the attached end for which the deflection is 14.5 inches. You ma elect to give students the following data set, which was actuall generated eperimentall ( and were both measured in inches). 4 8 12 16 20 24 28 32 0.5 1.75 3.75 6.5 9.5 13 16.75 20.5
Peterson, Technical Mathematics, 3rd edition 11 Student Worksheet 16.5 A Graphical Eploration of Rational Functions The object of this activit is for ou to make generalizations about how to find asmptotes and intercepts of rational functions. This activit should be done in groups of 2 3 students so several graphs can be generated simultaneousl b group members. Using a graphing utilit, graph each rational function and estimate the vertical asmptote, the horizontal asmptote, and the -intercept from the graph. After all functions have been graphed, look for patterns in the chart, draw several hpotheses pertaining to algebraic methods that would locate asmptotes and intercepts, and test our hpotheses with other rational functions of our choice. Function Vertical Asmptote Horizontal Asmptote -Intercept 2 + 1 3 + 2 3 + 6 3 3 2 4 + 1 2 5 + 4 2 4 2 5 + 4 2 2 + 8 10 2 + 9
Peterson, Technical Mathematics, 3rd edition 12 Student Worksheet 16.6 The Salt Tank A tank, initiall containing 40 gallons of a water-salt solution with 20 pounds of dissolved salt, has 2 gallons of another salt solution being pumped in each minute. The solution being pumped into the tank contains 1 pound of salt for ever 10 gallons of miture. 1. Complete the table below for the number of gallons of solution in the tank, the amount of salt in the tank, and the concentration of salt in the tank in pounds per gallon. 2. Then, b recognizing a pattern in the chart, find a smbolic model for the gallons, the pounds of salt, and the concentration. 3. Using the smbolic model, determine after how man minutes the concentration will reach (a) a 30% level? (b) a 20% level? (c) a 10% level? (d) a 5% level? 4. What level will the concentration approach as time increases indefinitel? Time Gallons of Solution Pounds of Salt Concentration 0 min 1 min 2 min 3 min 4 min t min
Answers Student Worksheet 16.1 1. A cubic polnomial has the two basic shapes as shown in Figures.6.1 and.6.2. Each of these basic shapes ma also be reflected in a vertical or a horizontal line. FIGURE.6.1 FIGURE.6.2 A cubic graph must cross the -ais at least once and so there is at least one real root. The curvature of the graph changes once from concave up to concave down, or vice versa. The point on the curve where the concavit changes is the inflection point. All cubic graphs have one inflection point. The -value of each cubic function goes toward positive infinit at one end of the graph (that is, as approaches positive or negative infinit) and toward negative infinit at the other end. 2. A fourth-degree (or quartic) polnomial has the three basic shapes as shown in Figures.6.3.6.5. Each of these basic shapes ma also be reflected in a vertical or a horizontal line. FIGURE.6.3 FIGURE.6.4 FIGURE.6.5 A fourth-degree graph does not have to cross the -ais. If it does cross the -ais then it will cross either two or four times. The curvature of the graph ma not change. That is, the entire curve ma be concave up, as in Figure.6.3 or it ma be concave down. It the curvature changes then it will be from concave up to concave down and then back to concave up, or from concave down to concave up and then back to concave down. The point on the curve where the concavit changes is the inflection point. All quartic graphs have either zero or two inflection points. The -value of each cubic function goes toward positive infinit at both ends of the graph (that is, as approaches positive or negative infinit) or toward negative infinit at both ends. 13
Instructional Resource Guide, Answers Peterson, Technical Mathematics, 3rd edition 14 Student Worksheet 16.2 1a. 1e. 1b. 1f. Cannot be done. 1g. 1c. 1h. Cannot be done. 1d. 2. Ever cubic is of the form f() = a( r 1 )( r 2 )( r 3 ) where r 1, r 2, and r 3 are the roots. This indicates that each cubic must have three roots. 3. Smmetr with respect to the -ais means that ever point on the curve has a mirror image across the -ais. This is the same as saing that f() = f() for ever value of. But, 3 ( ) 3 so a cubic function cannot have smmetr with respect to the -ais. Student Worksheet 16.3 2. about 0.448 d 3a. In 1986 3b. 1988 3c. Yes, 1991 4. 1.772 cm 5a. Yes 5b. Twice, at about 0.0031 and again at about 0.021. Student Worksheet 16.4 1. Answers will var depending on which degree of polnomial is used and the particular points being used. For eample, if a quadratic polnomial, a 2 +b+c = k, is desired and the points (90, 9820), (175, 9505), and (212, 9540) are used, then we get the following sstem of equations. 8100a + 90b + c = 9820 175 2 a + 175b + c = 9505 212 2 a + 212b + c = 9540
Instructional Resource Guide, Answers Peterson, Technical Mathematics, 3rd edition 15 The resulting function is f() 0.038129 2 13.810263 + 10754.07. Using this function, the heat rate will be a minimum when about 181.1 MW are generated. At that time the heat rate will be approimatel 9503.6 BTU/MWH. Student Worksheet 16.5 16. 1. 2 + 1 (a) (a) [ 6.3, 6.3] [ 4, 4] 2. [ 6.3, 6.3] [ 6, 6] (b) Vertical Asmptote: = 1 (c) Horizontal Asmptote: = 0 (d) -Intercept: None 3 + 2 5. (b) Vertical Asmptotes: = 2 and = 2 (c) Horizontal Asmptote: = 0 (d) -Intercept: = 3 + 1 2 5 + 4 (a) (a) [ 6.3, 6.3] [ 4, 4] [ 12.6, 12.6] [ 7.2, 7.2] (b) Vertical Asmptotes: = 1 and = 4 (b) Vertical Asmptote: = 2 (c) Horizontal Asmptote: = 0 (c) Horizontal Asmptote: = 1 (d) -Intercept: = 1 3. (d) -Intercept: = 3 3 + 6 3 6. 2 4 2 5 + 4 (a) (a) [ 12.6, 12.6, 1] [ 10, 20, 2] (b) Vertical Asmptote: = 3 (c) Horizontal Asmptote: = 3 (d) -Intercept: = 2 [ 12.6, 12.6] [ 7, 7] (b) Vertical Asmptotes: = 1 and = 4 (c) Horizontal Asmptote: = 1 (d) -Intercept: = 2 and = 2 4. 3 2 4 7. 2 2 + 8 10 2 + 9
Instructional Resource Guide, Answers Peterson, Technical Mathematics, 3rd edition 16 (a) (b) Vertical Asmptotes: = 3 and = 3 (c) Horizontal Asmptote: = 2 (d) -Intercept: = 5 and = 1 [ 12.6, 12.6] [ 7, 7] Student Worksheet 16.6 1. Time Gallons of Solution Pounds of Salt Concentration 0 min 40 20 50% 1 min 42 20.2 48.1% 2 min 44 20.4 46.4% 3 min 46 20.6 44.8% 4 min 48 20.8 43.3% t min 40 + 2t 20 + 0.2t 20 + 0.2t 40 + 2t 100% 2. See the last row of the table. 3a. 20 min 3b. 60 min 3c. Never 3d. Never 4. 10%