Math 140: Unit 6

Due Date: Wednesday October 23, 2002

Topics

Unit 6 continues work with polynomial and rational functions. The first part of the unit focuses on the problem of finding the zeros of a polynomial, which is of particular importance in mathematics. This has historically been a difficult problem, but it is now much easier with the use of computers and graphing calculators. The second part of the unit uses the work on zeros to help solve polynomial and rational inequalities. In this unit, it will be important to become very familiar and adept at using the graphing features of your calculator. In particular, you will need to be able to use your calculator to approximate the zeros of a function.

• Finding the zeros of a quadratic polynomial (A.6)
  • Complex numbers
  • Quadratic equations with a negative discriminant
• Finding zeros of polynomials
  • The Remainder and Factor Theorems (3.6)
  • Finding the number of possible positive and negative real zeros (Descartes' Rule of Signs) (3.6)
  • Finding bounds on real zeros (3.6)
  • Finding all possible rational zeros, and then all actual rational zeros (3.6)
  • Finding all exact real zeros (3.6)
  • Using the Intermediate Value Theorem to locate real zeros (3.6)
  • Approximating real zeros (3.6)
  • Complex zeros (3.7)
  • The Fundamental Theorem of Algebra (3.7)
  • Finding all exact real and complex zeros (3.7)
• Solving polynomial and rational inequalities (3.5)

Unit 6 will be followed by the Midterm Exam.

Study guidelines

• Section A.6: Complex Numbers; Quadratic Equations with a Negative Discriminant
  • Review the quadratic formula in Appendix A.5 before you begin this section.
  • Reading: Appendix A.6
  • Problems: A.6 #1, 5, 9, 11, 13, 15, 19, 21, 33, 37, 39, 43, 45, 49, 51, 53, 59, 61, 63, 65, 67, 69, 71, 73, 75
    
    
• Section 3.6: The Real Zeros of a Polynomial Function
  • Review Appendix sections A.3-5 before you begin this section. In sections 3.6 and 3.7, you will need to be able to accurately divide polynomials using long division and/or synthetic division.
  • Reading: Chapter 3.6
    Pay particular attention to the procedure for finding exact real zeros outlined on page 199, along with example 6. Try to follow this procedure when you work through the exercises. You will also need to use this procedure in section 3.7.
  • In practice, it is most important to be able to approximate zeros using the graphing capabilities of your calculator. The "bisection method" outlined on page 203 and in example 10 is a more difficult method and much slower. Thus, while it is a good idea to understand the bisection method and work through a few problems using that method, you should concentrate on being able to find zeros using your calculator. See Appendix section B.3 for brief instructions, and see your calculator manual or Graphing Calculator Help for more detailed instructions.
  • Learn the "Strategy fo Finding Rational Zeros" on page 233.
  • Problems: 3.6 #1, 5, 9, 11, 15, 19, 23-33 odds, 35, 39, 43, 47, 51, 55, 59, 63, 71, 75, 79, 85, 87, 89, 91, 105, 106
    
    
• Section 3.7: Complex Zeros; Fundamental Theorem of Algebra
  • Reading: Chapter 3.7
    Example 3 is a good example of the complete procedure for finding all exact real and complex zeros of a polynomial function. Try to follow this procedure when you work through the exercises.
  • Problems: 3.7 #1-33 odds, 35-38
    
    
• Section 3.5: Polynomial and Rational Inequalities
  • Review Appendix A.8 before you begin this section.
  • Reading: Chapter 3.5
    Pay particular attention to the solution procedure outlined on pages 186-187. Each of the examples uses this procedure.Try to follow this procedure when you work through the exercises.
  • Problems: 3.5 #1-57 odds
    
    
• Supplementary material (optional) for Chapter 3:
  • Student Solutions Manual for Chapter 3 (see note)
  • Video lectures are available in the Media Reserve room of the ISU library. They are listed under Math 140: Poon.
  • For tutoring help, visit the Prentice Hall Tutor Center. Tutors can be contacted by phone, fax, or e-mail. To register, you will need the access code that came with your textbook.
  • Graphing Calculator Help
  • The MathPak5 web site contains other resources such as Reading Quizzes and links to other interesting and fun math sites.

Unit Exam #6
After completing the above work, do the following:

1. Read the exam description:
   • This exam has 15 questions, and will count 15 points toward your grade. 
   • The exam has a one hour time limit.
   • You will need your graphing calculator on this exam. In particular, you will need to know how to use your graphing calculator to graph polynomials and to approximate the zeros of polynomials (as in some of the questions in section 3.6).
   • You should also be able to do synthetic division quickly and accurately.
   • Be sure to look under the entry box for the expected format of the answer.
     • Some problems ask you to solve a problem which may have more than one solution. If there is more than one solution, separate the solutions with a semicolon. For example, your answer might be 3;5.
     • Some of the questions on this exam ask for an approximation instead of an exact answer. These questions will ask you to give an answer that is "accurate to at least two decimal places" (or possibly one decimal, or three decimals, etc.). In this case, you would have to enter at least two digits after the decimal point. For example, if you calculate your answer to be 1.41421356237, then you could enter 1.41, 1.414, 1.4142, etc. When you view your corrected exam, you will see that the "correct" answer is given in the form 1.4142 ? 0.01. The question mark means "plus or minus", so anything within 0.01 of the first number counts as correct.
     • If a question does not ask for an approximation, then your answer must be exact.
2. Complete the online Unit 6 Pretest assignment. You may use your book if you wish, and redo the pretest as many times as you like. Your pretest score will be scaled to 5 points maximum.
3. You can repeat the online Unit 6 Pretest assignment without using your book until you feel that you are ready for the unit exam.
4. If you are having trouble with any of the problems listed above or on the pretest or practice exams, spend some time working with the online tutors listed on the Help page.
5. Go to the Math Computer Lab (449 Carver) to take the online proctored unit exam (go to Unit 6 Exam). Remember to bring identification, and remember that you will not be able to take the unit exam after the deadline date given at the top of this page. You may NOT use your book or notes on this exam.

   Note: Remember that you can always go back and take practice exams (go to Unit 6 Practice Exam) on this unit after the deadline has passed. In particular, this will help you prepare for the midterm and final exams.

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