1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1 Homework, Page 234 Write the polynomial in standard form, and identify the zeros of the function and the x-intercepts of its graph. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 2 Homework, Page 234 Write a polynomial function in minimum degree in standard form with real coefficients whose zeros include those listed. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 3 Homework, Page 234 Write a polynomial function in minimum degree in standard form with real coefficients whose zeros include those listed. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 4 Homework, Page 234 Write a polynomial function in minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 5 Homework, Page 234 Match the polynomial function graph to the given zeros and multiplicities. 17. b.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 6 Homework, Page 234 State how many complex and real zeroes the function has. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 7 Homework, Page 234 State how many complex and real zeroes the function has. 25. From the graph, f has four complex zeros, two of which are real.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 8 Homework, Page 234 Find all zeros and write a linear factorization of the function. 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 9 Homework, Page 234 Using the given zero, find all of the zeros and write a linear factorization of f (x). 33.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 10 Homework, Page 234 Write the function as a product of linear and irreducible quadratic factors all with real coefficients. 37.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 11 Homework, Page 234 Write the function as a product of linear and irreducible quadratic factors all with real coefficients. 41.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 12 Homework, Page 234 Answer Yes or No. If yes, give an example; if no, give a reason. 45. Is it possible to find a polynomial of degree 3 with real number coefficients that has -2 as its only real zero. Yes, for example

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 13 Homework, Page 234 Find the unique polynomial with real coefficients that meets these conditions. 49. Degree 4; zeros at x = 3, x = –1, and x = 2 – i; f (0) = 30

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 14 Homework, Page 234 53. There is at least one polynomial with real coefficients with 1 – 2i as its only nonreal zero. Justify your answer. False, to have real coefficients, the nonreal factors of a polynomial must be in conjugate pairs.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 15 Homework, Page 234 57. Which of the following cannot be the number of nonreal zeros of a polynomial of degree 5 with real coefficients? a. 0 b. 2 c. 3 d. 4 e. None of the above.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 16 Homework, Page 234 61. Verify that the complex number i is a zero of the polynomial

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 17 Homework, Page 234 65. Find the three cube roots of 8 by solving

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.6 Graphs of Rational Functions

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 19 Quick Review

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 20 Quick Review Solutions

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 21 What you’ll learn about Rational Functions Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions … and why Rational functions are used in calculus and in scientific applications such as inverse proportions.

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 22 Rational Functions

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 23 Limits We used limits to investigate continuity in Chapter 1 Limits may also be used to investigate behavior near vertical asymptotes to investigate behavior as functions approach positive or negative infinity, usually called end behavior

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 24 Transformations of the Reciprocal Function The general form for a function is In this equation, k indicates units of vertical translation, h indicates units of horizontal translation, and a indicates factor of stretch. For instance, indicates the reciprocal function is translated 4 units left and 3 down and stretched by a factor of 3.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 25 Example Finding the Domain of a Rational Function

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 26 Graph a Rational Function

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 27 Graph a Rational Function (Cont’d)

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 28 Example Finding Asymptotes of Rational Functions

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 29 Example Graphing a Rational Function

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 30 Analyzing Graphs Use limits to describe the behavior of the function

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 31 Finding Properties of Graphs Find the intercepts, vertical asymptotes, end behavior asymptotes, and graph the function.

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 32 Homework Review Section: 2.6 Page 245, Exercises: 1 – 69 (EOO)

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Solving Equations in One Variable

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 34 Quick Review

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 35 Quick Review Solutions

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 36 What you’ll learn about Solving Rational Equations Extraneous Solutions Applications … and why Applications involving rational functions as models often require that an equation involving fractions be solved.

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 37 Extraneous Solutions When we multiply or divide an equation by an expression containing variables, the resulting equation may have solutions that are not solutions of the original equation. These are extraneous solutions. For this reason we must check each solution of the resulting equation in the original equation.

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 38 Example Solving by Clearing Fractions

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 39 Example Solving a Rational Function

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 40 Example Eliminating Extraneous Solutions

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 41 Example Eliminating Extraneous Solutions

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 42 Example Finding an Acid Solution

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 43 Example Finding an Acid Solution

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 44 Example Finding an Acid solution

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 45 Example Finding an Acid Solution

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 46 Example Finding a Minimum Perimeter

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 47 Example Finding a Minimum Perimeter