1. What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials Simplify rational expressions means that we could not reduce or factor anything else out of the expression. Simplify, Multiply, and Divide Rational Expressions

2. Now let’s reduce Polynomials When dividing polynomials, they are called ____rational expressions____ There are two steps for reducing/simplifying rational expressions. Step 1. ___Factor the numerator and denominator___ Step 2. _____Reduce/Cancel like terms.

3. Example 1-1a Simplify Look for common factors. Simplify. Answer:

4. Example 1-1b Under what conditions is this expression undefined? A rational expression is undefined if the denominator equals zero. To find out when this expression is undefined, completely factor the denominator. Answer: The values that would make the denominator equal to 0 are –7, 3, and –3. So the expression is undefined at y = –7, y = 3, and y = – 3. These values are called excluded values.

5. Example 1-1c a.Simplify b.Under what conditions is this expression undefined? Answer: Answer: undefined for x = –5, x = 4, x = –4

6. Example 1-2a Multiple-Choice Test Item For what values of p isundefined? A 5 B –3, 5 C 3, –5 D 5, 1, –3 Read the Test Item You want to determine which values of p make the denominator equal to 0.

7. Example 1-2b Solve the Test Item Look at the possible answers. Notice that the p term and the constant term are both negative, so there will be one positive solution and one negative solution. Therefore, you can eliminate choices A and D. Factor the denominator. Factor the denominator. Solve each equation. Answer:B Zero Product Property or

8. Example 1-2c Multiple-Choice Test Item For what values of p isundefined? A –5, –3, –2 B –5 C 5 D –5, –3 Answer:D

9. Example 1-3a Simplify Simplify. Answer:or –a

10. Example 1-3b Simplify Answer: –x

11. Example 1-4a Simplify Answer:Simplify.

12. Example 1-4b Simplify Answer:Simplify.

13. Example 1-4c Simplify each expression. a. b. Answer:

14. Example 1-5a Simplify Answer:Simplify.

15. Example 1-5b Simplify Answer:

16. Example 1-6a Simplify Answer:Simplify.

17. Example 1-6b Simplify Simplify.Answer:

18. Stopped here after day 1

19. Example 1-6c Answer: 1 Simplify each expression. a. b. Answer:

20. Example 1-7a Simplify Express as a division expression. Multiply by the reciprocal of divisor.

21. Example 1-7b Factor. 11–1 1 11 Simplify.Answer:

22. Example 1-7c Simplify Answer:

23. End of Lesson 1

24. Lesson 2 Contents Example 1LCM of Monomials Example 2LCM of Polynomials Example 3Monomial Denominators Example 4Polynomial Denominators Example 5Simplify Complex Fractions Example 6Use a Complex Fraction to Solve a Problem

25. Example 2-1a Find the LCM of 15a 2 bc 3, 16b 5 c 2, and 20a 3 c 6. Factor the first monomial. Factor the second monomial. Factor the third monomial.

26. Example 2-1b Use each factor the greatest number of times it appears as a factor and simplify. Answer:

27. Example 2-1c Find the LCM of 6x 2 zy 3, 9x 3 y 2 z 2, and 4x 2 z. Answer: 36x 3 y 3 z 2

28. Example 2-2a Find the LCM of x 3 – x 2 – 2x and x 2 – 4x + 4. Factor the first polynomial. Factor the second polynomial. Answer: Use each factor the greatest number of times it appears as a factor.

29. Example 2-2b Find the LCM of x 3 + 2x 2 – 3x and x 2 + 6x + 9. Answer:

30. Example 2-3a Simplify The LCD is 42a 2 b 2. Find equivalent fractions that have this denominator. Simplify each numerator and denominator. Add the numerators. Answer:

31. Example 2-3b Simplify Answer:

32. Example 2-4a Simplify Factor the denominators. The LCD is 6(x – 5). Subtract the numerators.

33. Example 2-4b Distributive Property Combine like terms. Simplify. 1 1 Answer:

34. Example 2-4c Simplify Answer:

35. Example 2-5a Simplify The LCD of the numerator is ab. The LCD of the denominator is b.

36. Example 2-5b Simplify the numerator and denominator. Write as a division expression. Multiply by the reciprocal of the divisor. 1 1

37. Example 2-5c Simplify.Answer:

38. Example 2-5d Simplify Answer:

39. Example 2-6a Coordinate Geometry Find the slope of the line that passes throughand Definition of slope

40. Example 2-6b The LCD of the numerator is 3k. The LCD of the denominator is 2k. Write as a division expression. Simplify. Answer: The slope is

41. Example 2-6c Coordinate Geometry Find the slope of the line that passes throughand Answer:

42. End of Lesson 2

44. Lesson 3 Contents Example 1Vertical Asymptotes and Point Discontinuity Example 2Graph with a Vertical Asymptote Example 3Graph with Point Discontinuity Example 4Use Graphs of Rational Functions

45. Example 3-1a Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of First factor the numerator and denominator of the rational expression.

46. Example 3-1b Answer: The function is undefined for x = –2 and –3. Since x = –3 is a vertical asymptote and x = –2 is a hole in the graph. 1 1

47. Example 3-1c Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of Answer: vertical asymptote: x = –5 ; hole: x = –3

48. Answer: Example 3-2a Graph The function is undefined for x = –1. Since is in its simplest form, x = –1 is a vertical asymptote. Draw the vertical asymptote.

49. Example 3-2b Make a table of values. xf (x)f (x) –41.33 –3 1.5 –2 2 0 0 1 0.5 20.67 30.75 Answer: Plot the points and draw the graph.

50. Example 3-2c As |x| increases, it appears that the y values of the function get closer and closer to 1. The line with the equation f (x) = 1 is a horizontal asymptote of the function. Answer:

51. Example 3-2d Graph Answer:

52. Example 3-3a Graph Notice thatorTherefore, the graph of is the graph of with a hole at

53. Example 3-3b Answer:

54. Example 3-3c Graph Answer:

55. Example 3-4a Transportation A train travels at one velocity V 1 for a given amount of time t 1 and then another velocity V 2 for a different amount of time t 2. The average velocity is given by Let t 1 be the independent variable and let V be the dependent variable. Draw the graph if V 1 = 50 miles per hour, V 2 = 30 miles per hour, and t 2 = 1 hour.

56. Example 3-4b Answer: The function is The vertical asymptote is Graph the vertical asymptote and the function. Notice that the horizontal asymptote is

57. Example 3-4c What is the V -intercept of the graph? Answer:The V -intercept is 30.

58. Example 3-4d What values of t 1 and V are meaningful in the context of the problem? Answer:In the problem context, time and velocity are positive values. Therefore, positive values of t 1 and V values between 30 and 60 are meaningful.

59. Example 3-4e Transportation A train travels at one velocity V 1 for a given amount of time t 1 and then another velocity V 2 for a different amount of time t 2. The average velocity is given by a.Let t 1 be the independent variable and let V be the dependent variable. Draw the graph if V 1 = 60 miles per hour, V 2 = 30 miles per hour, and t 2 = 1 hour.

60. Example 3-4f Answer:

61. b. What is the V -intercept of the graph? c.What values of t 1 and V are meaningful in the context of the problem? Example 3-4g Answer:The V -intercept is 30. Answer: t 1 is positive and V is between 30 and 60.