Math 140: Unit 6

Target Date: Friday Oct. 23
Deadline: Thursday Nov. 5

Note: The information on this page is for the 8th edition of the textbook.
Click here for the 7th edition information.
Click here for the 6th edition information.

Table of Contents:

Topics
Study Guidelines
Unit Pretest and Exam Description
Checklist

Topics

Unit 4 continues work with polynomial and rational functions. The first part of the unit uses the work on graphing in sections 4.1-4 to help solve polynomial and rational inequalities. However, the methods shown only work if you are able to factor the particular polynomial in question (or the numerator and denomator of the rational function). This is equivalent to being able to find the zeros of the polynomial. So the second part of the unit focuses on the general problem of finding the zeros of a polynomial. This problem is of particular importance in mathematics. It has historically been a difficult problem, but it is now much easier with the use of computers and graphing calculators. 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.

• Solving polynomial and rational inequalities (3.5 & 4.4)
• Finding exact rational and real zeros of polynomials
  • The Remainder and Factor Theorems (4.5)
  • Finding the number of possible positive and negative real zeros (Descartes' Rule of Signs) (4.5)
  • Finding bounds on real zeros (4.5)
  • Finding all possible rational zeros, and then all actual rational zeros (4.5)
  • Finding all exact real zeros (4.5)
  • Using the Intermediate Value Theorem to locate real zeros (4.5)
• Approximating real zeros (4.5)
• Finding all exact real and complex zeros of polynomials (A.7 & 4.6)
  • Complex numbers (A.7)
  • Quadratic equations with a negative discriminant (A.7)
  • Complex zeros (4.6)
  • The Fundamental Theorem of Algebra (4.6)
  • Finding all exact real and complex zeros (4.6)

Unit 6 will be followed by the Midterm Exam.

Study Guidelines for the 8th edition of Sullivan's Precalculus

These reading and problem assignments are designed to help you learn the course material. You should complete all of these problems, check your answers in the back of the textbook, and get help with the problems that you missed. Most of the problems are odd-numbered, so you can check the solutions in the Solutions Manual.

The only way to learn mathematics is to do mathematics, so while these problems will not be collected or graded, you will probably not do well in the course if you do not complete these and check your work as described above. After completing these problems, go on to the Unit Exam Description below and follow directions.

• Section 3.5: Quadratic Inequalities
  • Review appendix A.9 before you begin this section.
  • Reading: section 3.5
    Read and work through examples 1-2 and their matched problems.
  • Practice Problems: 3.5 #1, 2, 3-35 odds
    
    
• Section 4.4: Polynomial and Rational Inequalities
  • Reading: section 4.4
    Read and work through examples 1-4 and their matched problems.
  • Pay particular attention to the solution procedure outlined on pages 210. Each of the examples uses this procedure.Try to follow this procedure when you work through the exercises.
  • Practice Problems: 4.4 #1-51 odds
    
    
• Section 4.5: The Real Zeros of a Polynomial Function
  • Review appendix sections A.3-4 before you begin this section. In sections 4.5 and 4.6, you will need to be able to accurately divide polynomials using long division and/or synthetic division.
  • Reading: section 4.5
    Read and work through examples 1-10 and their matched problems.
  • Learn the strategy for finding rational zeros via the Rational Zeros Theorem (exhibited in examples 4 and 5).
  • Pay particular attention to the procedure for finding exact real zeros outlined on page 221, 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 4.6.
  • In practice, it is most important to be able to approximate zeros using the graphing capabilities of your calculator. The "Intermediate Value Theorem method" outlined on page 225 and in example 10 is a more difficult method and much slower. Thus, while it is a good idea to understand the Intermediate Value Theorem 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.
  • Practice Problems: 4.5 #1-4, 11, 15, 19, 21, 25, 29, 33-43 odds, 45, 49, 53, 57, 61, 65, 69, 73, 81, 85, 89, 95, 97, 99, 101, 111, 112
    
    
• Section A.7: Complex Numbers; Quadratic Equations with a Negative Discriminant
  • Review the quadratic formula in Appendix A.6 before you begin this section.
  • Reading: appendix section A.7
    Read and work through examples 1-12 and their matched problems.
  • Practice Problems: A.7 #9, 13, 17, 19, 21, 23, 27, 29, 41, 45, 47, 51, 53, 57, 59, 61, 67, 69, 71, 73, 75, 77, 79, 81, 83
    
    
• Section 4.6: Complex Zeros; Fundamental Theorem of Algebra
  • Reading: section 4.6
    Read and work through examples 1-3 and their matched problems.
  • 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.
  • Additional discussion and examples of the procedure for finding zeros of polynomials.
  • Practice Problems: 4.6 #1, 2, 7-39 odds, 41-44
    
    
• Supplementary material (optional):
  • Student Solutions Manual
  • Algebra Review booklet
  • CD lecture series (step-by-step video examples on CD)
  • 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

Unit 6 Pretest and Exam Description

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 4.5).
   • 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 symbol 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 and notes if you wish.
   • Directions: Click on the link above, then choose the Unit 6 Pretest.
   • The pretest must be completed by the deadline date listed at the top of this page.
     However, you may redo the pretest as many times as you like before the due date.
     Your best score counts, and it will be rescaled to 5 points maximum.
3. If you are having trouble with any of the problems listed above or on the pretest, make use of the help resources listed on the Help page.
4. Arrange with your proctor to take the online proctored 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.
   • Directions: Click on the link above, then choose Unit 6 Exam.
   • The proctored unit exam must be completed by the deadline date listed at the top of this page, and may be repeated under certain conditions. See the Detailed Schedule page for specific rules.

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

   • Directions: Click on the link above, then choose Unit 6 Practice Exam. After the deadline has passed, this exam will be available in practice mode.

Unit 6 Checklist

To finish Unit 6, make sure that you have completed the following items:

• Read the material listed in the Study Guidelines.
• Practice the problems listed in the Study Guidelines, and use any of the listed supplementary material to help you understand the concepts.
• Complete the online Unit Pretest (read the exam description first).
• Complete the online proctored Unit Exam.

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