Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

3.2E: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

Practice Makes Perfect

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

1. a. 2x2z b. 4p16p5 c. n3n2+2n8

Answer

a. z=0
b. p=56
c. n=4,n=2

2. a. 10m11n b. 6y+134y9 c. b8b236

3. a. 4x2y3y b. 3x22x+1 c. u1u23u28

Answer

a. y=0
b. x=12
c. u=4,u=7

4. a. 5pq29q b. 7a43a+5 c. 1x24

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

5. 4455

Answer

45

6. 5663

7. 8m3n12mn2

Answer

2m23n

8. 36v3w227vw3

9. 8n963n36

Answer

83

10. 12p2405p100

11. x2+4x5x22x+1

Answer

x+5x1

12. y2+3y4y26y+5

13. a24a2+6a16

Answer

a+2a+8

14. y22y3y29

15. p3+3p2+4p+12p2+p6

Answer

p2+4p2

16. x32x225x+50x225

17. 8b232b2b26b80

Answer

4b(b4)(b+5)(b8)

18. 5c210c10c2+30c+100

19. 3m2+30mn+75n24m2100n2

Answer

3(m+5n)4(m5n)

20. 5r2+30rs35s2r249s2

21. a55a

Answer

1

22. 5dd5

23. 205yy216

Answer

5y+4

24. 4v3264v2

25. w3+216w236

Answer

w26w+36w6

26. v3+125v225

27. z29z+2016z2

Answer

z54+z

28. a25z3681a2

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

29. 1216·410

Answer

310

30. 325·1624

31. 5x2y412xy3·6x220y2

Answer

x38y

32. 12a3bb2·2ab29b3

33. 5p2p25p36·p21610p

Answer

p(p4)2(p9)

34. 3q2q2+q6·q299q

35. 2y210yy2+10y+25·y+56y

Answer

y53(y+5)

36. z2+3zz23z4·z4z2

37. 284b3b3·b2+8b9b249

Answer

4(b+9)3(b+7)

38. 72m12m28m+32·m2+10m+24m236

39. c210c+25c225·c2+10c+253c214c5

Answer

c+53c+1

40. 2d2+d3d216·d28d+162d29d18

41. 2m23m22m2+7m+3·3m214m+153m2+17m20

Answer

(m3)(m2)(m+4)(m+3)

42. 2n23n1425n2·n210n+252n213n+21

Divide Rational Expressions

In the following exercises, divide the rational expressions.

43. v511v÷v225v11

Answer

1v+5

44. 10+ww8÷100w28w

45. 3s2s216÷s34s2+16ss364

Answer

3ss+4

46. r2915÷r3275r2+15r+45

47. p3+q33p2+3pq+3q2÷p2q212

Answer

4(p2pq+q2)(pq)(p2+pq+q2)

48. v38w32v2+4vw+8w2÷v24w24

49. x2+3x104x÷(2x2+20x+50)

Answer

x28x(x+5)

50. 2y210yz48z22y1÷(4y232yz)

51. 2a2a215a+20a2+7a+12a2+8a+16

Answer

2a75

52. 3b2+2b812b+183b2+2b82b27b15

53. 12c2122c23c+14c+46c213c+5

Answer

3(3c5)

54. 4d2+7d235d+10d247d212d4

For the following exercises, perform the indicated operations.

55. 10m2+80m3m9·m2+4m21m29m+20÷5m2+10m2m10

Answer

4(m+8)(m+7)3(m4)(m+2)

56. 4n2+32n3n+2·3n2n2n2+n30÷108n224nn+6

57. 12p2+3pp+3÷p2+2p63p2p12·p79p39p2

Answer

(4p+1)(p4)3p(p+9)(p1)

58. 6q+39q29q÷q2+14q+33q2+4q5·4q2+12q12q+6

Multiply and Divide Rational Functions

In the following exercises, find the domain of each function.

59. R(x)=x32x225x+50x225

Answer

x5 and x5

60. R(x)=x3+3x24x12x24

61. R(x)=3x2+15x6x2+6x36

Answer

x2 and x3

62. R(x)=8x232x2x26x80

For the following exercises, find R(x)=f(x)·g(x) where f(x) and g(x) are given.

63. f(x)=6x212xx2+7x18g(x)=x2813x227x

Answer

R(x)=2

64. f(x)=x22xx2+6x16g(x)=x264x28x

65. f(x)=4xx23x10g(x)=x2258x2

Answer

R(x)=x+52x(x+2)

66. f(x)=2x2+8xx29x+20g(x)=x5x2

For the following exercises, find R(x)=f(x)g(x) where f(x) and g(x) are given.

67. f(x)=27x23x21g(x)=3x2+18xx2+13x+42

Answer

R(x)=3x(x+7)x7

68. f(x)=24x22x8g(x)=4x3+28x2x2+11x+28

69. f(x)=16x24x+36g(x)=4x224xx2+4x45

Answer

R(x)=x(x5)x6

70. f(x)=24x22x4g(x)=12x2+36xx211x+18

Writing Exercises

71. Explain how you find the values of x for which the rational expression x2x20x24 is undefined.

Answer

Answers will vary.

72. Explain all the steps you take to simplify the rational expression p2+4p219p2.

73. a. Multiply 74·910 and explain all your steps.
b. Multiply nn3·9n+3 and explain all your steps.
c. Evaluate your answer to part b. when n=7. Did you get the same answer you got in part a.? Why or why not?

Answer

Answers will vary.

74. a. Divide 245÷6 and explain all your steps.
b. Divide x21x÷(x+1) and explain all your steps.
c. Evaluate your answer to part b. when x=5. Did you get the same answer you got in part a.? Why or why not?

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and six rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was determine the values for which a rational expression is undefined. In row 3, the I can was simplify rationale expressions. In row 4, the I can was multiply rational expressions. In row 5, the I can was divide rational expressions. In row 6, the I can was multiply and divide rational functions. There is the nothing in the other columns.

b. If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.


This page titled 3.2E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

Support Center

How can we help?