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Mathematics LibreTexts

9.3E: Exercises

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Practice Makes Perfect

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, add.

1. 215+715

Answer

35

2. 724+1124

3. 3c4c5+54c5

Answer

3c+54c5

4. 7m2m+n+42m+n

5. 2r22r1+15r82r1

Answer

r+8

6. 3s23s2+13s103s2

7. 2w2w216+8ww216

Answer

2ww4

8. 7x2x29+21xx29

In the following exercises, subtract.

9. 9a23a7493a7

Answer

3a+7

10. 25b25b6365b6

11. 3m26m3021m306m30

Answer

m22

12. 2n24n3218n164n32

13. 6p2+3p+4p2+4p55p2+p+7p2+4p5

Answer

p+3p+5

14. 5q2+3q9q2+6q+84q2+9q+7q2+6q+8

15. 5r2+7r33r2494r2+5r+30r249

Answer

r+9r+7

16. 7t2t4t2256t2+12t44t225

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

17. 10v2v1+2v+412v

Answer

4

18. 20w5w2+5w+625w

19. 10x2+16x78x3+2x2+3x138x

Answer

x+2

20. 6y2+2y113y7+3y23y+1773y

21. z2+6zz2253z+2025z2

Answer

z+4z5

22. a2+3aa293a279a2

23. 2b2+30b13b2492b25b849b2

Answer

4b3b7

24. c2+5c10c216c28c1016c2

Find the Least Common Denominator of Rational Expressions

In the following exercises, a. find the LCD for the given rational expressions b. rewrite them as equivalent rational expressions with the lowest common denominator.

25. 5x22x8,2xx2x12

Answer

a. (x+2)(x4)(x+3)
b. 5x+15(x+2)(x4)(x+3),
2x2+4x(x+2)(x4)(x+3)

26. 8y2+12y+35,3yy2+y42

27. 9z2+2z8,4zz24

Answer

a. (z2)(z+4)(z4)
b. 9z36(z2)(z+4)(z4),
4z28z(z2)(z+4)(z4)

28. 6a2+14a+45,5aa281

29. 4b2+6b+9,2bb22b15

Answer

a. (b+3)(b+3)(b5)
b. 4b20(b+3)(b+3)(b5),
2b2+6b(b+3)(b+3)(b5)

30. 5c24c+4,3cc27c+10

31. 23d2+14d5,5d3d219d+6

Answer

a. (d+5)(3d1)(d6)
b. 2d12(d+5)(3d1)(d6),
5d2+25d(d+5)(3d1)(d6)

32. 35m23m2,6m5m2+17m+6

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

33. 710x2y+415xy2

Answer

21y+8x30x2y2

34. 112a3b2+59a2b3

35. 3r+4+2r5

Answer

5r7(r+4)(r5)

36. 4s7+5s+3

37. 53w2+2w+1

Answer

11w+1(3w2)(w+1)

38. 42x+5+2x1

39. 2yy+3+3y1

Answer

2y2+y+9(y+3)(y1)

40. 3zz2+1z+5

41. 5ba2b2a2+2bb24

Answer

b(5b+10+2a2)a2(b2)(b+2)

42. 4cd+3c+1d29

43. 3m3m3+5mm2+3m4

Answer

mm+4

44. 84n+4+6n2n2

45. 3rr2+7r+6+9r2+4r+3

Answer

3(r2+6r+18)(r+1)(r+6)(r+3)

46. 2ss2+2s8+4s2+3s10

47. tt6t2t+6

Answer

2(7t6)(t6)(t+6)

48. x3x+6xx+3

49. 5aa+3a+2a+6

Answer

4a2+25a6(a+3)(a+6)

50. 3bb2b6b8

51. 6m+612mm236

Answer

6m6

52. 4n+48nn216

53. 9p17p24p21p+17p

Answer

p+2p+3

54. 13q8q2+2q24q+24q

55. 2r16r2+6r1652r

Answer

3r2

56. 2t30t2+6t2723t

57. 2x+710x1+3

Answer

4(8x+1)10x1

58. 8y45y+26

59. 3x23x42x25x+4

Answer

x5(x4)(x+1)(x1)

60. 4x26x+53x27x+10

61. 5x2+8x94x2+10x+9

Answer

1(x1)(x+1)

62. 32x2+5x+212x2+3x+1

63. 5aa2+9a2a+18a22a

Answer

5a2+7a36a(a2)

64. 2bb5+32b2b152b210b

65. cc+2+5c210cc24

Answer

c5c+2

66. 6dd5+1d+4+7d5d2d20

67. 3dd+2+4dd+8d2+2d

Answer

3(d+1)d+2

68. 2qq+5+3q313q+15q2+2q15

Add and Subtract Rational Functions

In the following exercises, find a. R(x)=f(x)+g(x) b. R(x)=f(x)g(x).

69. f(x)=5x5x2+x6 and g(x)=x+12x

Answer

a. R(x)=(x+8)(x+1)(x2)(x+3)
b. R(x)=x+1x+3

70. f(x)=4x24x2+x30 and g(x)=x+75x

71. f(x)=6xx264 and g(x)=3x8

Answer

a. R(x)=3(3x+8)(x8)(x+8)
b. R(x)=3x+8

72. f(x)=5x+7 and g(x)=10xx249

Writing Exercises

73. Donald thinks that 3x+4x is 72x. Is Donald correct? Explain.

Answer

Answers will vary.

74. Explain how you find the Least Common Denominator of x2+5x+4 and x216.

75. Felipe thinks 1x+1y is 2x+y.
a. Choose numerical values for x and y and evaluate 1x+1y.
b. Evaluate 2x+y for the same values of x and y you used in part a..
c. Explain why Felipe is wrong.
d. Find the correct expression for 1x+1y.

Answer

a. Answers will vary.
b. Answers will vary.
c. Answers will vary.
d. x+yx

76. Simplify the expression 4n2+6n+91n29 and explain all your steps.

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 add and subtract rational expressions with a common denominator. In row 3, the I can was add and subtract rational expressions with denominators that are opposites. In row 4, the I can find the least common denominator of rational expressions. In row 5, the I can was add and subtract rational expressions with unlike denominators. In row 6, the I can was add or subtract rational functions. There is the nothing in the other columns.

b. After reviewing this checklist, what will you do to become confident for all objectives?


This page titled 9.3E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Chau D Tran.

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