CHAPTER 9 In this chapter ou epand our knowledge of families of functions to include polnomial functions. As ou investigate the equation! graph connection for polnomials, ou will learn how to search for factors (which can help ou find -intercepts) and how to use division to find additional factors. When ou investigate the graphs of polnomials and sstems involving polnomials, ou will see man that appear not to intersect. As ou investigate these sstems further, ou will learn about imaginar and comple numbers. In the last section of the chapter, ou will appl our knowledge of polnomials to model some of the attractions at a count fair. Polnomials Think about these questions throughout this chapter:? How can I model this situation? How can I find the other solutions? How can I graph it? What is an imaginar number? In this chapter ou will learn: How to sketch complete graphs of polnomial functions. How to divide one polnomial epression b another. How to solve some equations ou could not solve earlier b epanding the set of numbers ou use. What imaginar and comple numbers are, their properties, and how the are related to graphs that do not have real intersections. How to find solutions for factorable polnomial equations of one variable. How to write polnomial equations given their graphs and how to use polnomial functions to represent some situations. Section 9.1 Section 9.2 Section 9.3 You will investigate polnomial functions and learn how to sketch them without using a graphing calculator. You will also learn how to find polnomial equations from their graphs. Here ou will be introduced to a number sstem called comple numbers. You will see how these numbers relate to quadratic equations that have no real solutions and to the roots of other polnomial equations. In this section, ou will learn how to divide polnomials b appling our knowledge of generic rectangles and factoring. You will solve equations using polnomial division to factor polnomials. 786 Algebra 2 Connections
Chapter 9 Teacher Guide Section Lesson Das Lesson Objectives Materials Homework 9.1 9.1.1 2 9.1.2 1 Sketching Graphs of Polnomial Functions More Graphs of Polnomials Poster graph paper Colored poster pens 9-8 to 9-17 and 9-18 to 9-27 None 9-38 to 9-46 9.1.3 1 9.2.1 1 Stretch Coefficients for Polnomial Functions Introducing Imaginar Numbers None 9-56 to 9-64 None 9-72 to 9-80 9.2 9.2.2 1 Comple Roots None 9-89 to 9-97 9.2.3 1 More Comple Numbers and Equations Lesson 9.2.3 Res. Pg. (optional) Scissors (optional) 9-105 to 9-113 9.3.1 1 Polnomial Division None 9-121 to 9-130 9.3 9.3.2 2 or 3 Factors and Integral Roots None 9-140 to 9-147 and 9-148 to 9-155 and 9-156 to 9-164 9.3.3 1 An Application of Polnomials Graph paper Scissors Tape 9-169 to 9-176 Chapter Closure Total: Varied Format Options 11 or 12 das plus optional time for Chapter Closure Overview of the Chapter In Section 9.1, students will investigate the equation! graph connections for polnomial functions. The will recognize that equations in factored form are much easier to sketch and the will understand the relationship between the factors and the -intercepts of the graph. Then, in Section 9.2, the will develop understanding of imaginar and comple numbers and recognize that polnomial functions can have comple roots. In Lesson 9.2.3, which introduces the comple plane, students can eplore one possible graphical representation of comple roots. In Section 9.3, the will learn to divide polnomials b a known factor to find other factors. This will allow them to find comple and irrational roots of some cubic and quartic functions. Algebra Strand This chapter will require students to appl their algebra skills, specificall adding, subtracting, multipling, and dividing polnomials in order to graph polnomial functions. Chapter 9: Polnomials 787
Lesson 9.1.1 How can I predict the graph? Sketching Graphs of Polnomial Functions Lesson Objective: Length of Activit: Core Problems: Was of Thinking: Materials: Suggested Lesson Activit: Students will describe the graph of a polnomial given its equation in factored form. Two das (approimatel 100 minutes) Problems 9-1 and 9-2 (with 9-3 through 9-6 if using Further Guidance ) Justifing, generalizing, investigating Poster graph paper and markers Da 1: Introduction: Read the introduction and lead a brief discussion about the polnomials students alread know. Then start teams on problem 9-1. The should recognize that the first two equations given have graphs that are parabolas. Figuring out how those parabolas might be related to the roller-coaster graph is more difficult, however. The third equation provides a clue that more factors ma help. If teams are struggling, ou can ask students to guess what kind of equations might have three or four -intercepts. The intent of this introductor eercise is for students to get used to guessing an equation and entering it on the graphing calculator to check it. Do not epect all teams to come up with great guesses or ver good fits before moving on to the investigation in problem 9-2. If teams do not have accurate guesses, ou can direct them to return to this problem after completing the rest of the lesson, at which point the should be able to find a better fit. Investigation (problem 9-2): Team members can initiall divide up the given polnomials and sketch graphs on their own graph paper, or if the are using the Further Guidance section, the should work on the first three together and then divide up the rest. Students should look for, label, and describe the -intercepts and bounces (double roots), the -intercept, the number of turns, and the behavior of graphs for ver large and ver small -values. The should share and discuss all of their graphs and their observations within their teams. Discussion should lead to conjectures and the creation of several new equations to tr. As ou circulate, listen to see if students are coming up with - and -intercepts, predictions about the numbers of crossings of the -ais, and ideas for determining a reasonable window or maimum and minimum approimations. If so, continue to circulate and listen. Encourage teams that have trouble getting started to think about multiple representations. The might find an table useful (particularl for the equations in factored form where the have the -intercepts and then can use numbers between those numbers to tr -values to investigate peaks and valles). For teams that are starting to make some conjectures about -ais crossings or turns, ou might ask, What if ou multipl b 788 Algebra 2 Connections
another factor? How will multipling b another factor affect the graph? How will it affect the equation? or What might the graph look like if the highest power of were five or si? If teams are going in a number of different directions, it will be useful to bring the whole class together for a progress report in which each one can share one idea the have come up with so far or what their strateg is. In an case, bring the class together at the end of the first da for a quick progress report or possibl to return to problem 9-1 to consider additional guesses. Da 2: While teams are developing their summar statements and posters, circulate and use vocabular relating to polnomials (such as degree, factors, zeros, coefficients, etc.) in the contet of our remarks. For instance, when a team comes up with the idea that when a polnomial has a factor of (! 5) there is an -intercept at (5, 0), point out that P(5) = 0 and that 5 is called a zero of the polnomial or a root of the equation. This should be casual at this point and not interrupt the flow of teams thinking. Students will focus more formall on the terminolog in Lesson 9.1.2. Presentations: Have each team present its findings to the class. During and after presentations, use and discuss the terms root, zero, degree, and polnomial. Ask teams to add this vocabular to their posters. Closure: (15 minutes) Lead a class discussion in which ou and the class make a list on the board or overhead of useful tools for graphing polnomial functions. Clarif definitions, if needed, as ou record elements in the list. Be sure the list includes ideas of factored form, -intercepts, the effect of repeated factors, and checking -values between intercepts as well as the terms root, zero, degree, and polnomial. Direct students to the learning log prompt in problem 9-7. If time permits, ask students to graph a few polnomials in factored form in the air. Students should use a finger to trace their visualization of the general shape of the graphs in the air as ou write them on the board or overhead. Of course their shapes will be backwards for ou, but ou can get a rough idea of P() = ( + 9)( + 6)(! 10) students level of understanding. A few Q() = ( + 5)( + 3)(! 4)(! 8) possible equations are R() = ( + 8)( + 3)(! 4)(! 9)(! 15) shown at right. S() =!( + 5)( + 2)(! 4)(! 10) Homework: Da 1: problems 9-8 through 9-17 Da 2: problems 9-18 through 9-27 Note: If after Da 1 ou anticipate needing three das to complete the lesson, ou can assign problems 9-8 through 9-14 on Da 1, problems 9-18 through 9-24 on Da 2 and use problems 9-15 through 9-17 and 9-25 through 9-27 for homework on Da 3. Chapter 9: Polnomials 789
9.1.1 How can I predict the graph? Sketching Graphs of Polnomial Functions In previous courses and chapters, ou learned how to graph man tpes of functions, including lines and parabolas. Toda ou will work with our team to appl what ou know to more complicated polnomial equations. Just as quadratic polnomial equations can be written in standard or factored form, other polnomial equations can be written in standard or factored form. For eample, = 4! 4 3! 3 2 + 10 + 8 is in standard form, and it can be written in factored form as = ( + 1) 2 (! 2)(! 4). During this lesson, ou will develop techniques for sketching the graph of a polnomial function from its equation, and ou will justif wh those techniques work. 9-1. The Mathamericaland Carnival Compan has decided to build a new roller coaster to use at this ear s count fair. The new coaster will have a ver special feature: part of the ride will be underground. The designers will use polnomial functions to describe different pieces of the track. Part of the design is shown at right. Your task is to guess a possible equation to represent the track and test it on our graphing calculator. To help get an idea of what to tr, start b checking the graphs of the equations given 1 2 3 below. Think about how the graphs are the 1 unit = 100 ft same and how the are different. = (! 2) = (! 2) 2 = (! 2)(! 3) Your task: Use the information ou found b graphing the above equations to help ou make guesses about the equation that would produce the graph of the roller coaster. Once ou have found a graph that has a shape close to this one, tr zooming in or changing the viewing window on our graphing calculator to see the details better. Keep track of what ou tried and the equations ou find that fit most accuratel. [ A reasonable guess would be = (! 2) 2 (! 3). ] 790 Algebra 2 Connections Height
9-2. POLYNOMIAL FUNCTION INVESTIGATION In this investigation, ou will determine which information in a polnomial equation can help ou sketch its graph. Your task: With our team, create summar statements eplaining the relationship between a polnomial equation and its graph. To accomplish this task, first divide up the equations listed below so each team member is responsible for two or three of them. Make a complete graph of each of our functions. Whenever possible, start b making a sketch of our graph without using a graphing calculator. Then, as a team, share our observations including our responses to the Discussion Points. Choose two or three equations that can be used to represent all of our findings. You can choose them from the list below, or ou can create new ones as a team. [ See graphs below. ] The form of our presentation to the class can be on a poster, transparencies or as a PowerPoint TM presentation. Whichever format our teacher decides, make sure ou include complete graphs and summar statements that are well justified. P 1 () = (! 2)( + 5) 2 P 2 () = 2(! 2)( + 2)(! 3) P 3 () = 4! 21 2 + 20 P 4 () = ( + 3) 2 ( + 1)(! 1)(! 5) P 5 () =!0.1( + 4) 3 P 6 () = 4! 9 2 P 7 () = 0.2( + 1)(! 3)( + 4) P 8 () = 4! 4 3! 3 2 + 10 + 8 Chapter 9: Polnomials 791
What can we predict from looking at the equation of a polnomial? Wh does this make sense? Which form of a polnomial equation is most useful for making a graph? What information does it give? How can we use the equation to help predict what a useful window might be? Which eamples are most helpful in finding the connections between the equation and the graph? How does changing the eponent on one of the factors change the graph? 9-3. As a team, eamine the first polnomial P 1 () = (! 2)( + 5) 2. [ a: cubics; when the factors are multiplied there is an 3, b: Eample: The graph goes upwards toward 5, bounces downward at 5 then turns upward again to go through = 2, then continues upward and is ver steep. ] a. What famil of functions is it a member of? How do ou know? Based on its equation, sketch the shape of its graph. b. Now use our graphing calculator to graph P 1 (). Label the -intercepts. How are the -intercepts related to the equation? Reading from left to right along the -ais, describe the graph before the first -intercept, between -intercepts, and after the last -intercept. 9-4. Continuing as a team, eamine the equation P 2 () = 2(! 2)( + 2)(! 3). [ a: There are three factors; -intercepts are ( 2, 0), (2, 0), and (3, 0); Eample: The shapes are similar, but this graph has three intercepts and the graph in part (a) crosses once and then bounces off the -ais. b: No; it affects the stretch and the -intercept; the negative flips the graph. ] a. How man distinct (different) factors are there? How man -intercepts would ou predict for its graph? Draw the graph and label the -intercepts. How is this graph similar to or different from the graph of P 1 ()? b. Does the factor 2 have an effect on the -intercepts? On the shape of the graph? On the -intercepts? How would the graph change if the factor 2 were changed to be a factor 2? 792 Algebra 2 Connections
9-5 What is different about P 3 () = 4! 21 2 + 20? What -intercept(s) can ou determine from the equation, before graphing with the calculator? Eplain how ou know. Use the graph to figure out eactl what the other intercepts are. Eplain how ou can prove that our answers are eact. [ It is not factored; (0, 0) because ou can factor out an or ou can see that when = 0, then = 0; the trace button will give an approimation, choose the closest integer and substitute to check. ] 9-6. With our team, divide up the work to investigate P 4 () through P 8 () and continue our investigation, referring back to the our task statement and the discussion points in problem 9-2. Further Guidance section ends here. 9-7. Based on what our team learned and on the class discussion, record our own list of useful strategies for graphing polnomial functions. Use as man of the new vocabular words as ou can and write down the ones ou are not sure of et. You will add to and refine this list over the net several lessons. ETHODS AND MEANINGS MATH NOTES Polnomials, Degree, Coefficients A polnomial in one variable is an epression that can be written as the sum or difference of terms of the form: an!number ( ) ( )! whole!number Polnomials with one variable (often ) are usuall arranged with powers of in order, starting with the highest, left to right. Polnomials can include onl the operations of addition, subtraction, or multiplication. The highest power of the variable in a polnomial of one variable is called the degree of the polnomial. The numbers that multipl each term are called coefficients. See the eamples below. Eample 1: f () = 7 5 + 2.5 3! 1 + 7 is a polnomial function of degree 5 2 with coefficients 7, 0, 2.5, 0,! 1, and 7. Note that the last term, 7, is called the 2 constant term but represents the variable epression 7 0, since 0 = 1. Eample 2: = 2( + 2)( + 5) is a polnomial in factored form with degree 2 because it can be written in standard form as = 2 2 + 14 + 20. It has coefficients 2, 14, and 20. The following are not polnomial functions: = 2! 3, f () = 1 2!2 +, and =! 2. Chapter 9: Polnomials 793
9-8. For each equation below, make tables that include -values from 2 to 2 and draw each graph. [ See graphs below. Parent functions: a: = 3, b: = 4, c: = 3 ] a. = (! 1) 2 ( + 1) b. = (! 1) 2 ( + 1) 2 c. = 3! 4 d. What are the parent functions for these equations? 9-9. Polnomials are epressions that can be written as a sum of terms of the form: (an number)! (whole number) Which of the following equations are polnomial equations? For those that are not polnomials, eplain wh not. Check the lesson 9.1.1 Math Notes bo for further details about polnomials. [ Functions in parts (a), (b), and (e) are polnomial functions; eplanations var. ] a. f () = 8 5 + 2 + 6.5 4 + 6 b. = 3 5 6 +19 2 c. = 2 + 8 d. f () = 9 +! 3 e. P() = 7(! 3)( + 5) 2 f. = 2 + 1 2 +5 g. Write an equation for a new polnomial function and then write an equation for a new function that is not a polnomial. 9-10. Describe the possible numbers of intersections for each of the following pairs of graphs. Sketch a graph for each possibilit. For eample, a circle could intersect a line twice, once, or not at all. Your solution to each part should include all of the possibilities and a sketched eample of each one. [ a: 0, 1, or! ; b: 0, 1, or 2; c: 0, 1, 2, 3, or 4; d: 0, 1, 2, 3, or 4 (1 and 3 require the parabola to be tangent to the circle.) ] a. Two different lines. b. A line and a parabola. c. Two different parabolas. d. A parabola and a circle. 794 Algebra 2 Connections
9-11. Solve the following sstem: = 2! 5 [ ( 2, 1) and (3, 4) ] = +1 9-12. A table can be used as a useful tool for finding some inverse functions. When the function has onl one in it, the function can be described with a sequence of operations, each applied to the previous result. Consider the following table for f () = 2! 1 + 3. 1 st 2 nd 3 rd 4 th What f does to : subtracts 1 multiplies b 2 adds 3 Since the inverse must undo these operations, in the opposite order, the table for f!1 () would look like the one below. 1 st 2 nd 3 rd 4 th What does f!1 to : subtracts 3 divides b 2 (!!!!) 2 adds 1 a. Cop and complete the following table for g!1 () if g() = 1 3 ( + 1)2! 2 1 st 2 nd 3 rd 4 th What g does to : adds 1 (!!!) 2 divides b 3 subtracts 2 What g!1 does to : adds 2 multiplies b 3 subtracts 1 b. Write the equations for f!1 () and g!1 (). [ f!1 () = (! 3 2 ) 2 + 1, g!1 () = 3( + 2)! 1 ] 9-13. Describe the difference between the graphs of = 3! and = 3! + 5. [ The second graph is shifted up 5 from the first. ] 9-14. Solve the equations below. [ a: =!26, b: = 10 or = 3 ] a. 3 +2 + 7!2 = 3 b.!7!5 = 6 9-15. An arithmetic sequence starts out!23,!!19,!!15 [ a: 4n! 27, b: at least 2507 times ] a. What is the rule? b. How man times must the generator be applied so that the result is greater than 10,000? Chapter 9: Polnomials 795
9-16. Artemis was putting up the sign at the Count Fair Theater for the movie ELVIS RETURNS FROM MARS. He got all of the letters he would need and put them in a bo. He reached into the bo and pulled out a letter at random. [ a: 20 2 = 10 1 ; b: 19 1 ] a. What is the probabilit that he got the first letter he needed when he reached into the bo? b. Once he put the first letter up, what is the probabilit that he got the second letter he needed when he reached into the bo? 9-17. Without a calculator, find two solutions 0!! " < 360! that make each of the following equations true. [ a: 60, 300 ; b: 135, 315 ; c: 60, 120 ; d: 150, 210 ] a. cos! = 1 2 b. tan! = " 2 2 c. sin! = 3 2 d. cos! = " 3 2 9-18. Which of the following equations are polnomial functions? For each one that is not, justif wh not. [ The functions in parts (a), (b), (d), (e), (h), (i), and (j) are polnomial functions. ] a. = 3 2 + 2 2 + b. = (!1) 2 (! 2) 2 c. = 2 + 2 d. = 3!1 e. = (! 2) 2!1 f. 2 = (! 2) 2!1 g. = 1 2 + 1 + 1 2 h. = 1 2 + 1 3 i. = j. =!7 9-19. Samantha thinks that the equation (! 4) 2 + (! 3) 2 = 25 is equivalent to the equation (! 4) + (! 3) = 5. Is she correct? Are the two equations equivalent? Eplain how ou know. If the are not equivalent, eplain Samantha s mistake. [ The are not equivalent. Eplanations var. Students ma substitute numbers to check. Also, the second equation can be written =! + 12, which is a line, not a circle. ] 796 Algebra 2 Connections
9-20. Find the roots (the solutions when = 0) of each of the following polnomial functions. [ a: = 2 or = 4, b: = 3, c: =!2, = 0, or = 2 ] a. = 2! 6 + 8 b. f () = 2! 6 + 9 c. = 3! 4 9-21. Sketch a graph of = 2! 7. [ See graph at right. a: 2, b: = 7,!! 7 ] a. How man roots does this graph have? b. What are the roots of the function? 9-22. Solve 2 + 2! 5 = 0. [ =!1 ± 6, a: 2, b: at! 1.45 and! "3.45 ] a. How man -intercepts does = 2 + 2! 5 have? b. Approimatel where does the graph of = 2 + 2! 5 cross the -ais? 9-23. This is a checkpoint for finding the equation for the inverse of a function. Consider the function f () = 2 3(! 1) + 5. [ f!1 () = 1 3 (! 2 5 ) 2 + 1 = 12 1 (! 5) 2 + 1 for! 5, see graph at right. ] a. Find the equation for the inverse of f (). b. Sketch the graph of both the original and the inverse. c. Check our results b referring to the Checkpoint 17 materials located at the back of our book. If ou needed help to write the equation of the inverse of this function, then ou need more practice with inverses. Review the Checkpoint 17 materials and tr the practice problems. Also, consider getting help outside of class time. From this point on, ou will be epected to write equations for inverses of functions such as this one quickl and easil. 9-24. Graph the inequalit 2 + 2! 25, and then describe its graph in words. [ See graph at right. ] Chapter 9: Polnomials 797
9-25. Find if 2 p() = 4 and p() = 2! 4! 3. [ 1 or 5 ] 9-26. Start with the graph of = 3, then write new equations that will shift the graph as described below. [ a: = (3 )! 4, b: = 3 (! 7) ] a. Down 4 units. b. Right 7 units. 9-27. THE COUNTY FAIR FERRIS WHEEL Consider this picture of a Ferris wheel. The wheel has a 60-foot diameter and is drawn on a set of aes with the Ferris wheel s hub (center) at the origin. Use a table like the one below and draw a graph that relates the angle (in standard position) of the spoke leading to our seat to the approimate height of the top of our seat above or below the height of the central hub. The table below starts at 90, our starting position before ou ride around the wheel. [ a: Repeat the pattern for several ccles, b: 30, c: = 30 sin ] (angle) 90 45 0 45 90 135 180... 270 (height) 30' 21.2' 0' 21.2' 30' 21.2' 0' 30' a. The wheel goes around (counter-clockwise) several times during a ride. How could ou reflect this fact in our graph? Update our graph. b. What is the maimum distance above or below the center that the top of our seat attains during the ride? c. Find an equation to fit the Count Fair Ferris wheel ride. 798 Algebra 2 Connections
Lesson 9.1.2 How can I predict the graph? More Graphs of Polnomials Lesson Objective: Length of Activit: Students will consolidate, generalize, and eplain their findings from the polnomial investigation. One da (approimatel 50 minutes) Core Problems: Problems 9-29 through 9-37 Was of Thinking: Materials: Materials Preparation: Suggested Lesson Activit: Closure: (10 minutes) Justifing, generalizing None Write equations to graph in the air on the board or overhead. Start toda with several graphs for students to sketch in the air. Then, if needed, review as a whole class the list of what students know so far based on the Polnomial Function Investigation. You can use the equations in problem 9-28, as well as these additional possibilities: f () = ( + 7)( + 2)( 1)( 5), g() = ( + 4)( 1)( 6), h() = ( + 2) 2 ( 3)( 5), and j() = ( + 2) 3. Teams should work on problems 9-29 through 9-37, discussing each question and coming to an agreement. As ou circulate, listen for the use of mathematical terms such as equation, degree of a polnomial, factors, roots, and intercepts. As ou ask teams to eplain their thinking, encourage them to use these terms. Consider using a Participation Quiz to highlight successful use of such vocabular. Note: In problem 9-30, the phrase negative orientation is used to refer to the fact that the stretch coefficient is negative. The resulting graphs all have f () becoming ver small (or ver negative ) as the value of gets ver large. The intent here is for students to understand that as the -value grows, the -value gets ver small (or ver negative ). Students ma describe these as graphs where, The right arrow goes down. Ask several students to put their number lines from problem 9-36 on the board or overhead and challenge the class to come up with the graphs and equations. Use these eamples to discuss generalizations about the graphs of polnomial functions based on their degree and number of distinct or repeated factors. Homework: Problems 9-38 through 9-46 Chapter 9: Polnomials 799
9.1.2 How can I predict the graph? More Graphs of Polnomials Toda ou will use what ou learned in the Polnomial Function Investigation to respond to some questions. Thinking about how to answer these questions should help ou clarif and epand on some of our ideas as well as help ou learn how to use the vocabular involved with polnomials. 9-28. Use our finger to trace an approimate graph of polnomial functions in the air, as directed b our teacher. Or ou ma sketch each of the polnomial functions below quickl on paper. Just sketch the graph without the - and - aes. [ See sketches below. ] a. P() = ( + 10)( + 7)(! 12) b. Q() = ( + 6)( + 3)(! 5)(! 8) c. R() =!( + 4)( + 2)(! 6)(! 10) d. W () = ( + 7) 2 (! 7) 2 e. S() = ( + 6)( + 3)(! 5)(! 8)(! 12) 800 Algebra 2 Connections
9-29. Look back at the work ou did in Lesson 9.1.1 (problem 9-2, Polnomial Function Investigation ). Then answer the following questions. [ a: 3; b: n; c: Yes, if an factors are repeated. At this point, students ma also sa when it is shifted above the -ais (because it has comple roots). ] a. What is the maimum number of roots a polnomial of degree 3 can have? Sketch an eample. b. What do ou think is the maimum number of roots a polnomial of degree n can have? c. Can a polnomial of degree n have fewer than n roots? Under what conditions? 9-30. For each polnomial function shown below, state the minimum degree its equation could have. [ (i): 3, (ii): 2, (iii): 4, (iv): 5; a: Graphs (i) and (iii); b: As the -values get larger the -values continue to decrease. c: Graphs (i) and (iii) have a positive orientation and graphs (ii) and (iv) have a negative orientation. ] i. ii. iii. iv. a. Which of the graphs above show that as the -values get ver large the -values continue to get larger and larger? b. How would ou describe the other graphs for ver large -values? c. When the -values of a graph get ver large as the -values get large, the graph has positive orientation. When the -values of a graph get ver small as the -values get large, the graph has negative orientation. How is each of the above graphs oriented? Chapter 9: Polnomials 801
9-31. For each graph in problem 9-30, ou decided what the minimum degree of its equation could be. Under what circumstances could graphs that look the same as these have polnomial equations of a higher degree? [ a: The have the same general shape, but = (! 1) 4 flattens out more and rises more quickl. b: Yes; c: No, it would have to turn again. d: The have the same general shape, but = 5 grows more quickl, so it appears narrower. e: Polnomials that have all different factors have humps. These just bend. ] Consider the graphs of = (! 1) 2 and = (! 1) 4. a. How are these graphs similar? How are the different? b. Could the equation for graph (ii) from the previous problem have degree 4? c. Could it have degree 5? Eplain. d. How is the graph of = 3 similar to or different from the graph of = 5? e. How do the shapes of graphs of = (! 2) 3 and = ( + 1) 5 differ from the shapes of graphs of equations that have three or five factors that are all different? 9-32. In P 1 () = (! 2)( + 5) 2 (the first eample from the Polnomial Function Investigation ), ( + 5) 2 is a factor. This produces what is called a double root of the function. [ a: It causes the graph to be tangent at the root (sometimes said to bounce off ). b: The graph flattens out as it passes through the -ais at the root. ] a. What effect does this have on the graph? b. Check our equations for a triple root. What effect does a triple root have on the graph? 9-33. We can use a number line to represent the -values for which a polnomial graph is above or below the -ais. The bold parts of each number line below show where the output values of a polnomial function are positive (that is, where the graph is above the -ais). The open circles show locations of the -intercepts or roots of the function. Where there is no shading, the value of the function is negative. Sketch a possible graph to fit each number line, and then write a possible equation. (Each number line represents the -ais for a different polnomial.) [ a: = ( + 3)( + 1)(! 4), b: =!( 2 )( + 5)(! 2), c: = (! 1) 2 ( + 1)(! 4), d: =!( + 2)( + 1)(! 1)(! 2) ] a. b. -3-1 4 c. d. -1 1 4-5 0 2-2 -1 1 2 802 Algebra 2 Connections
9-34. What can ou sa about the graphs of polnomial functions with an even degree compared to the graphs of polnomial functions with an odd degree? Use graphs from the Polnomial Functions Investigation (and mabe some others), to justif our response. [ Polnomial functions with an even degree start and end from the same direction. Polnomial functions with an odd degree start and end from opposite directions. ] 9-35. Choose three of the polnomials ou graphed in the Polnomial Functions Investigation (problem 9-2) and create number lines for their graphs similar to the ones in problem 9-33. [ answer number lines below ] P 1 () = (! 2)( + 5) 2 P 2 () = 2(! 2)( + 2)(! 3) P 3 () = 4! 21 2 + 20 P 4 () = ( + 3) 2 ( + 1)(! 1)(! 5) P 5 () =!0.1( + 4) 3 P 6 () = 4! 9 2 P 7 () = 0.2( + 1)(! 3)( + 4) P 8 () = 4! 4 3! 3 2 + 10 + 8 5 0 2 4 0 2 2 3 5 1 4 0 3 0 3 3 1 1 5 4 1 0 3 1 2 4 9-36. Create a new number-line description (like the ones in problem 9-33) and then trade with a partner. (Each team member should create a different number line.) After ou have traded, find a possible graph and equation for a polnomial function to fit the description ou have received. Then justif our results to our team and check our team members results. 9-37. Without using a calculator, sketch rough graphs of the following functions. [ See graphs below. ] a. P() =!( +1)(! 3) b. P() = (!1) 2 ( + 2)(! 4) c. P() = ( + 2) 3 (! 4) Chapter 9: Polnomials 803
ETHODS AND MEANINGS Roots and Zeros MATH NOTES The roots of a polnomial function, p(), are the solutions of the equation p() = 0. Another name for the roots of a function is zeros of the function because at each root, the value of the function is zero. The real roots (or zeros) of a function have the same value as the -values of the -intercepts of its graph because the -intercepts are the points where the -value of the function is zero. Sometimes roots can be found b factoring. In the Parabola Lab (problems 4-13 and 4-14), ou discovered how to make a parabola sit on the -ais (the polnomial has one root), and ou looked at was of making parabolas intersect the -ais in two specific places (two roots). 9-38. Where does the graph = ( + 3) 2! 5 cross the -ais? [ at (!3 ± 5, 0) ] 9-39. If ou were to graph the function f () = (! 74) 2 ( + 29), where would the graph intersect the -ais? [ at (74, 0), a double root, and at ( 29, 0) ] 9-40. For each pair of intercepts given below, write an equation for a quadratic function in standard form. [ possible answers: a: = 2!! 6, b: = 2 2 + 5! 3 ] a. (!3, 0) and (2, 0) b. (!3, 0) and ( 1 2, 0) 9-41. What is the degree of each polnomial function below? [ a: 2, b: 5, c: 3, d: 6 ] a. P() = 0.08 2 + 28 b. = 8 2! 1 7 5 + 9 c. f () = 5( + 3)(! 2)( + 7) d. = (! 3) 2 ( +1)( 3 +1) 804 Algebra 2 Connections
9-42. Are parabolas polnomial functions? Are lines polnomial functions? Are cubics? Eponentials? Circles? In all cases, eplain wh or wh not. [ Yes, es, es, no, no. Eplanations var. ] 9-43. A sequence of pentagonal numbers is started at right. [ a: 13, 17, and 21; b: arithmetic; c: 4n! 3 ] a. Find the net three pentagonal numbers. b. What kind of sequence do the pentagonal numbers form? 1 5 9 c. What is the equation for the n th pentagonal number? 9-44. A circle with its center on the line = 3 in the 1 st quadrant is tangent to the -ais. [ a: (! 2) 2 + (! 6) 2 = 4, b: (! 3) 2 + (! 9) 2 = 9 ] a. If the radius is 2, what is the equation of the circle? b. If the radius is 3, what is the equation of the circle? 9-45. Sketch the graph of each function below on the same set of aes. [ See graph at right. ] a. = 2 b. = 2 + 5 c. = 2! 5 9-46. For each equation, find two solutions 0!! " < 360!, which make the equation true. You should not need a calculator. [ a: 30, 150 ; b: 60, 240 ; c: 30, 330 ; d: 225, 315 ] a. sin! = 1 2 c. cos! = 3 2 b. tan! = 3 d. sin! = " 2 2 Chapter 9: Polnomials 805
Lesson 9.1.3 How can I find the equation? Stretch Coefficients for Polnomial Functions Lesson Objective: Students will write eact equations for the graphs of polnomial functions given the -intercepts and one additional point. Length of Activit: One da (approimatel 50 minutes) Core Problems: Problems 9-49 through 9-55 Was of Thinking: Materials: Suggested Lesson Activit: Generalizing, choosing a strateg None Start b asking the class a question such as, You have been writing possible equations for the graphs of polnomials, but how precisel do our equations fit the points on the graphs? How can ou tell how accurate the are? Solicit ideas and then start teams working on problems 9-47 through 9-49, which will bring up the inaccurac of the -values and the as-of-et undetermined stretch coefficient. Note that in problem 9-47, it is epected that students will generate equations from roots and the will plug in points, which the ma need to estimate, in order to check their equations. Those the come up with for parts (a) and (b) are likel to be accurate, but the will probabl not guess the correct stretch factor for part (c). Instead, students should notice that, while the can construct an equation from roots, the still need to find a wa to make it fit the graph eactl. This motivates the net part of the lesson. As ou circulate, ask teams to eplain their thinking. The most difficult jump for some teams ma be deciding that substitution of the coordinates of a point on the graph can help determine the value of a in the last part of problem 9-49. If students have difficult with this idea, ask them if the can recall a similar situation where the knew everthing ecept the stretch coefficient. The should recall finding the value of a for parabolas and other parent functions. Problems 9-52 through 9-55 raise the question of how man points are needed to determine third-degree polnomial equations. Some students ma remember using the three points with the standard form of the quadratic equation = a 2 + b + c to form and solve three linear equations in three variables in order to determine the equation of a parabola. The ma epand on that to suggest four points for a cubic, using the equation = a 3 + b 2 + c + d. However, other students ma come up with different eplanations, such as the intercepts plus one more. Be sure to leave time for a whole-class discussion that deals with which four points are needed and whether the lead to four independent equations. 806 Algebra 2 Connections
Closure: (15 minutes) Problems 9-54 and 9-55 raise a question that should be discussed with the whole class. Put a sketch of a graph of a third-degree polnomial equation with a double root on the board or overhead, with the -intercepts clearl marked, no scale on the -ais, and one other point ((1, 4) if ou use the eample suggested below) and ask the class to outline a method for determining its polnomial equation. After students have outlined a method and come up with an equation, ask, Could there be a different possible equation for this graph? Have ou made an assumptions about the equation based on what ou see? If there is a different possible equation, what could it be? If not, how could ou verif that the one we have is correct? A good eample to use might be = 1 3 (! 3)2 ( + 2) and then compare it to = 1 3 (! 3)4 ( + 2). Students could predict that a fifth-degree polnomial equation might be possible. If there is another known point to check, the question ma be resolved. (It will be resolved unless the -value leaves ±1 in the repeated factor.) Note that the points (4, 2) and (2, 4 ) work in both equations, but the 3 -intercepts differ: (0, 6) versus (0, 54). The purpose of this is for students to decide that the can substitute the coordinates of a point on the graph into the equation to check the fit. Have students share their ideas about what information about a graph is needed to determine the equation. Homework: Problems 9-56 through 9-64 Chapter 9: Polnomials 807
9.1.3 How can I find the equation? Stretch Coefficients for Polnomial Functions In Lesson 9.1.2 ou found possible equations for the graphs of polnomial functions based on their -intercepts. Man of the sketches ou used did not even include the scale on the -ais. In this lesson, ou will focus on figuring out equations that represent all of the points on the graphs. 9-47. Find reasonable equations for each of the following polnomial functions. Without using a graphing calculator, how can ou check the accurac of our equations? How can ou check to see whether the -values (or stretch factor) are accurate? Show how ou checked the accurac in each case. Were our equations accurate? [ a: = ( + 3)(! 2), ma use =!2 or = 1 to check; b: =! 3 ( + 3)(! 2), ma use =!2, =!1 or = 1 to check; c: Students are likel to tr = ( + 2) 2 (! 1) or =!( + 2) 2 (! 1) and find that it does not check. ] a. b. c. 9-48. What is the difference between the graphs of the functions = 2 (! 3)( + 1) and = 3 2 (! 3)( + 1)? [ The second graph is a vertical stretch of the first. ] 808 Algebra 2 Connections
9-49. ARE THE INTERCEPTS ENOUGH? Melvin wrote the equation = ( + 3)( + 1)(! 2) 2 to represent the graph at right. How well does this equation represent the graph? [ a: test a point, = 2( + 3)( + 1)(! 2) 2 ; b: You could substitute the coordinates into the equation for and and solve for a. a = 2 You could plug in other points to see how well the fit. ] a. Eplain how ou can decide how well the equation represents the graph. What can ou do to the equation to make it a better fit for the graph? What equation would fit better? b. Before ou figured it out, ou could have written the polnomial for this graph as P() = a( + 3)( + 1)(! 2) 2. What if ou did not have a graphing calculator, but ou were told that the graph goes through the point (1, 16)? How could ou use that information to determine the eact equation? Once ou have decided on a method with our team, tr it. How can ou test the accurac of our equation? 9-50. THE COUNTY FAIR COASTER RIDE Now that ou have more epertise with polnomial equations and their graphs, the Mathamericaland Carnival Compan has hired our team to find the eact equation to represent its roller-coaster track. The numbers along the -ais are in hundreds of feet. At 250 feet, the track will be 20 feet below the surface. This gives the point (2.5,!0.2). [ a: min: 4 th -degree; b: 0, 2, and 3. 2 is a double root; c: = 0.64(! 2) 2 (! 3) ; d: About 181 feet ] a. What degree polnomial represents the portion of the roller coaster represented b the graph at right? b. What are the roots? Height c. Find an eact equation for the polnomial that will generate the curve of the track. d. What is the deepest point of the roller coaster s tunnel? Chapter 9: Polnomials 809
9-51. Write an eact equation for each graph below. [ a: =!2( + 2) 2 (! 2), b: =! 3 4 ( + 2) 2 (! 1) 2 ] a. b. 9-52. Write a polnomial equation for a function with a graph that bounces off the -ais at (!1, 0), crosses it at (4, 0), and goes through the point ( 2, 18). [ A likel answer is = 3( + 1) 2 (! 4), but other answers are possible. ] 9-53. Armando came up with the equation = 3( + 1) 4 (! 4) for problem 9-52. Does his equation fit all of the given criteria? Wh or wh not? Is it the same as the equation ou came up with? [ Yes; the bounce is accounted for and substitution verifies that the given points work in the equation. Answers var. ] 9-54. What if problem 9-52 also had said that the graph went through the point (1,!36)? Is there still more than one possible equation? Eplain. [ No, because the new point onl satisfies the equation = 3( + 1) 2 (! 4). ] 9-55. What information about the graph of a polnomial function is necessar to determine eactl one correct equation? Discuss this with our team. [ Possible sample answers: ou have to know how man times a factor is used, or ou need an additional point to check. ] 810 Algebra 2 Connections
MATH NOTES OOKING DEEPER Notation for Polnomials A general equation to represent all polnomials: The general equation of a second-degree (quadratic) polnomial is often written in the form f () = a 2 + b + c, and the general equation of a third-degree (cubic) polnomial is often written in the form f () = a 3 + b 2 + c + d. For a polnomial with an undetermined degree n, it is unknown how man letters will be needed for the coefficients, so instead of using a, b, c, d, e, etc., mathematicians use onl the letter a, and the put on subscripts, as shown below. f () = (a n ) n + (a n!1 ) (n!1) +... + (a 1 ) 1 + a 0 This general polnomial has degree n and coefficients a n, a n!1,..., a 1, a 0. For eample, for 7 4! 5 3 + 3 2 + 7 + 8, the degree is 4. In this specific case, a n is a 4 and a 4 = 7, a n!1 is a 3 =!5, a n!2 is a 2 = 3, a 1 = 7, and a 0 = 8. 9-56. What is the stretch factor (the value of a) for the equation of the graph in part (c) of problem 9-47? Write the eact equation of the function. [ Stretch factor is!2. f () =!2( + 2) 2 (! 1) ] 9-57. For each of the following polnomial epressions, find the degree, list the coefficients, and then label them a 0 through a n. Refer to the eample in the Lesson 9.1.3 Math Notes bo above about polnomial notation. [ a: degree 4, a 4 = 6, a 3 =!3, a 2 = 5, a 1 = 1, a 0 = 8 ; b: degree 3, a 3 =!5, a 2 = 10, a 1 = 0, a 0 = 8 ; c: degree 2, a 2 =!1, a 1 = 1, a 0 = 0 ; d: degree 3, a 3 = 1, a 2 =!8, a 1 = 15, a 0 = 0 ; e: degree 1, a 1 = 1 ; f: degree 0, a 0 = 10 ] a. 6 4! 3 3 + 5 2 + + 8 b.!5 3 + 10 2 + 8! c.! 2 +! d. (! 3)(! 5) e. f. 10 Chapter 9: Polnomials 811
9-58. Write a polnomial equation for a graph that has three -intercepts: ( 4, 0), (1, 0), and (3, 0), and passes through the point ( 1, 60). [ possible equation: p() = 2.5( + 4)(! 1)(! 3) ] 9-59. The -intercepts of a quadratic polnomial are given below. Find a possible quadratic equation in standard form. [ Possible answers: a: = 4 2 + 5! 6, b: = 2! 5 ] a. = 3,! =!2 b. =! 5,! = 5 4 9-60. Consider the functions = 1 and = 16 2 the functions intersect. [ (±6, 1 2) ] 2!4. Find the coordinates where the graphs of 9-61. Find the center and radius of each circle below. [ a: C: (3, 7), r: 5; b: C: (0, 5), r: 4; c: C: ( 9, 4), r: 5 2 ; d: C: (3, 0), r: 1 ] a. (! 7) 2 = 25! (! 3) 2 b. 2 + 2 + 10 =!9 c. 2 + 2 + 18! 8 + 47 = 0 d. 2 + (! 3) 2 = 1 9-62. Without using a calculator, write the solution to each equation. [ a: log17 log 2, b: 242, c: 4, d: 7 ] a. 2 = 17 b. log 3 ( + 1) = 5 c. log 3 (3 ) = 4 d. 4 log 4 () = 7 9-63. Consider the function = 2 + 5 + 7. [ a: ( + 5 2) 2 + 3 4, verte (! 5 2, 3 4 ); b: (0, 7); c: ( 5, 7), see graph at right. ] a. Complete the square to find the verte. b. Find the -intercept. c. Use the verte, the -intercept, and the smmetr of parabolas to find a third point and sketch the graph. 9-64. Write a possible equation for the graph at right. [ = 2 + 4sin ] 812 Algebra 2 Connections
Lesson 9.2.1 What are imaginar numbers? Introducing Imaginar Numbers Lesson Objective: Students will solve equations using imaginar and comple numbers. Length of Activit: One da (approimatel 50 minutes) Core Problems: Problems 9-65 through 9-70 Was of Thinking: Materials: Suggested Lesson Activit: Generalizing None Start the lesson with teams working on problems 9-65 and 9-66 and reading the Math Notes boes that follow them. This could be either a whole-class reading and discussion, or a team effort followed b a whole class discussion to introduce imaginar numbers before the go ahead with problems 9-67 through 9-70. Problem 9-67 is an opportunit to practice in order to get used to the definition of i, before using it in contet. Note: There is an order of operations question that ma (or ma not) come up as teams are working on problem 9-67. When dealing with a numerical epression such as!7 the first step is to appl the definition to rewrite the epression as ( 7)i!or!i 7. For eample, consider the following problem:!7(!7) =? Appling the definition of i first leads to (i 7)(i 7) = i 2 49 =!7, which is the correct simplification. On the other hand, multipling first leads to 49 = 7.!7 must be defined using imaginar numbers before other operations can be performed. This result means that the general rule a! b = ab that holds for real numbers a! 0 and b! 0, cannot be etended to negative values for a and b. Wait until students raise this question to discuss it. This issue ma or ma not come up in this lesson. In an case, it should be discussed when students raise the question, which ma happen in this chapter or in a later chapter. After problem 9-67 teams should be able to use what the know about solving sstems of equations, solving quadratic equations, and operations with real numbers to complete problems 9-68 through 9-70, and, if time permits, problem 9-71 with their teams. Circulate and listen as the raise questions about imaginar and comple numbers. In preparation for future lessons, ou might ask teams the following questions while circulating: What do ou notice about the two solutions? What part of the Quadratic Formula determines whether there will be real or comple solutions? How could ou determine, in advance, what kind of solutions a quadratic equation will have? Chapter 9: Polnomials 813
Note: Students ma find it interesting that Bombelli s teacher, Cardano, was an earl accepter of negative numbers, but he referred to them as numeri ficti (fictional numbers). Closure: (10 minutes) This is a good time to discuss students questions about these new numbers. Refer back to the historical development of imaginar numbers and discuss their importance to modern mathematicians abilit to solve ever quadratic equation. Ask students whether the think the number sstem is now complete. Students ma have further questions about imaginar numbers, such as: Wh are the imaginar? and What good are the? You can refer back to their use in Bombelli s case, where he needed to use them in the process of solving a real problem. Imaginar numbers are used in some areas of phsics and electrical engineering in order to solve equations that lead back to real equations that solve real problems. Comple numbers (a + bi) were also a first step in thinking about multidimensional vectors, such as quaternions (a + bi + cj + dk), which also turn out to be useful in phsics. Homework: Problems 9-72 through 9-80 9.2.1 What are imaginar numbers? Introducing Imaginar Numbers In the past, ou have not been able to solve some quadratic equations like 2 + 4 = 0 and 2 + 1 = 0, because there are no real numbers ou can square to get a negative answer. To solve this issue, mathematicians created a new, epanded number sstem based on one new number. But this was not the first time mathematicians had invented new numbers! To read about other such inventions, refer to the Math Notes bo that follows problem 9-65. In this lesson, ou will learn about imaginar numbers and how ou can use them to solve equations ou were previousl unable to solve. 9-65. Consider the equation 2 = 2. [ a: square root, b: 2, c: 2, d: ± 2,! ± 1.4142 ] a. How do ou undo squaring a number? b. When ou solve 2 = 2, how man solutions should ou get? c. How man -intercepts does the graph of = 2! 2 have? d. Solve the equation 2 = 2. Write our solutions both as radicals and as decimal approimations. 814 Algebra 2 Connections
MATH NOTES OOKING DEEPER Historical Note: Irrational Numbers In Ancient Greece, people believed that all numbers could be written as fractions of whole numbers (what are now called rational numbers). Man individuals realized later that some numbers could not be written as fractions (such as 2 ), and these individuals challenged the accepted beliefs. Some of the people who challenged the beliefs were eiled or outright killed over these challenges. The Greeks knew that for a one-unit square, the length of the diagonal, squared, ielded 2. When it was shown that no rational number could do that, the eistence of what are called irrational numbers was accepted and smbols like 2 were invented to represent them. The problem 2 = 3 also has no rational solutions; fractions can never work eactl. The rational (i.e., decimal) solutions that calculators and computers provide are onl approimations; the eact answer can onl be represented in radical form, namel, ± 3. 9-66. Mathematicians throughout histor have resisted the idea that some equations ma not be solvable. Still, it makes sense that 2 + 1 = 0 cannot be solved because the graph of = 2 + 1 has no -intercepts. What happens when ou tr to solve 2 + 1 = 0? [ You need to take the square root of a negative number. ] Chapter 9: Polnomials 815
MATH NOTES OOKING DEEPER Historical Note: Imaginar Numbers In some was, each person s math education parallels the histor of mathematical discover. When ou were much ounger, if ou were asked, How man times does 3 go into 8? or What is 8 divided b 3? ou might have said, 3 doesn t go into 8. Then ou learned about numbers other than whole numbers, and the question had an answer. Of course, in some situations ou are onl interested in whole numbers, and then the first answer is still the right one. Later, if ou were asked, What number squared makes 5? ou might have said, No number squared makes 5. Then ou learned about numbers other than rational numbers, and ou could answer that question. Similarl, until about 500 ears ago, the answer to the question, What number squared makes 1? was, No number squared makes 1. Then something remarkable happened. An Italian mathematician named Bombelli used a formula for finding the roots of third-degree polnomials. Within the formula was a square root, and when he applied the formula to a particular equation, the number under the square root came out negative. Instead of giving up, he had a brilliant idea. He had alread figured out that the equation had a solution, so he decided to see what would happen if he pretended that there was a number he could square to make a negative. Remarkabl, he was able to continue the calculation, and eventuall the imaginar number disappeared from the solution. More importantl, the resulting answer worked; it solved his original equation. This led to the acceptance of these so-called imaginar numbers. The name stuck, and mathematicians became convinced that all quadratic equations do have solutions. Of course, in some situations ou will onl be interested in real numbers (that is, numbers not having an imaginar part), and then the original answer, that there is no solution, is still the correct one. 9-67. In the 1500s, an Italian mathematician named Rafael Bombelli invented the imaginar number!1, which is now called i.!1 = i implies that i 2 =!1. After this invention, it became possible to find solutions for 2 + 1 = 0 ; the are i and!i. What would be the value of!16 = 16(!1) = 16i 2 =? Use the definition of i to rewrite each of the following epressions. [ 4i, a: 2i, b: 6i 2 =!6, c: 4i 2 =! 4,! 4(!5i) = 20i, d: 5i ] a.! 4 b. (2i)(3i) c. (2i) 2 (!5i) d.!25 816 Algebra 2 Connections