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Mathematics LibreTexts

5.2E: Exercises

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Practice Makes Perfect

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Also give the degree of each polynomial.

1. ⓐ 47x517x2y3+y2
5c3+11c2c8
59ab+13b
4
4pq+17

Answer

ⓐ trinomial, degree 5
ⓑ other polynomial, degree 3
ⓒ binomial, degree 2
ⓓ monomial, degree 0
ⓔ binomial, degree 2

2. ⓐ x2y2
13c4
a2+2ab7b2
4x2y23xy+8
19

3. ⓐ 8y5x
y25yz6z2
y38y2+2y16
81ab424a2b2+3b
18

Answer

ⓐ binomial, degree 1
ⓑ trinomial, degree 2
ⓒ other polynomial, degree 3
ⓓ trinomial, degree 5
ⓔ monomial, degree 0

4. ⓐ 11y2
73
6x23xy+4x2y+y2
4y2+17z2
5c3+11c2c8

5. ⓐ 5a2+12ab7b2
18xy2z
5x+2
y38y2+2y16
24

Answer

ⓐ trinomial, degree 2
ⓑ monomial, degree 4
ⓒ binomial, degree 1
ⓓ other polynomial, degree 3
ⓔ monomial, degree 0

6. ⓐ 9y310y2+2y6
12p3q
a2+9ab+18b2
20x2y210a2b2+30
17

7. ⓐ 14s29t
z25z6
y38y2z+2yz216z3
23ab214
3

Answer

ⓐ binomial, degree 1
ⓑ trinomial, degree 2
ⓒ other polynomial, degree 3
ⓓ binomial, degree 3
ⓔ monomial, degree 0

8. ⓐ 15xy
15
6x23xy+4x2y+y2
10p9q
m4+4m3+6m2+4m+1

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

9. ⓐ 7x2+5x2
4a9a

Answer

12x25a

10. ⓐ 4y3+6y3
y5y

11. ⓐ 12w+18w
7x2y(12x2y)

Answer

6w
19x2y

12. ⓐ 3m+9m
15yz2(8yz2)

13. 7x2+5x2+4a9a

Answer

12x25a

14. 4y3+6y3y5y

15. 12w+18w+7x2y(12x2y)

Answer

6w+19x2y

16. 3m+9m+15yz2(8yz2)

17. ⓐ 5b17b
3xy(8xy)+5xy

Answer

22b
16xy

18. ⓐ 10x35x
17mn2(9mn2)+3mn2

19. ⓐ 12a+5b22a
pq24p3q2

Answer

10a+5b
pq24p3q2

20. ⓐ 14x3y13x
a2b4a5ab2

21. ⓐ 2a2+b26a2
x2y3x+7xy2

Answer

4a2+b2
x2y3x+7xy2

22. ⓐ 5u2+4v26u2
12a+8b

23. ⓐ xy25x5y2
19y+5z

Answer

xy25x5y2
19y+5z

24. 12a+5b22a+pq24p3q2

25. 14x3y13x+a2b4a5ab2

Answer

x3y+a2b4a5ab2

26. 2a2+b26a2+x2y3x+7xy2

27. 5u2+4v26u2+12a+8b

Answer

u2+4v2+12a+8b

28. xy25x5y2+19y+5z

29. Add: 4a,3b,8a

Answer

4a3b

30. Add: 4x,3y,3x

31. Subtract 5x6 from 12x6

Answer

7x6

32. Subtract 2p4 from 7p4

In the following exercises, add the polynomials.

33. (5y2+12y+4)+(6y28y+7)

Answer

11y2+4y+11

34. (4y2+10y+3)+(8y26y+5)

35. (x2+6x+8)+(4x2+11x9)

Answer

3x2+17x1

36. (y2+9y+4)+(2y25y1)

37. (8x25x+2)+(3x2+3)

Answer

11x25x+5

38. (7x29x+2)+(6x24)

39. (5a2+8)+(a24a9)

Answer

6a24a1

40. (p26p18)+(2p2+11)

In the following exercises, subtract the polynomials.

41. (4m26m3)(2m2+m7)

Answer

2m27m+4

42. (3b24b+1)(5b2b2)

43. (a2+8a+5)(a23a+2)

Answer

11a+3

44. (b27b+5)(b22b+9)

45. (12s215s)(s9)

Answer

12s214s+9

46. (10r220r)(r8)

In the following exercises, subtract the polynomials.

47. Subtract (9x2+2) from (12x2x+6)

Answer

3x2x+4

48. Subtract (5y2y+12) from (10y28y20)

49. Subtract (7w24w+2) from (8w2w+6)

Answer

w2+3w+4

50. Subtract (5x2x+12) from (9x26x20)

In the following exercises, find the difference of the polynomials.

51. Find the difference of (w2+w42) and (w210w+24)

Answer

11w64

52. Find the difference of (z23z18) and (z2+5z20)

In the following exercises, add the polynomials.

53. (7x22xy+6y2)+(3x25xy)

Answer

10x27xy+6y2

54. (5x24xy3y2)+(2x27xy)

55. (7m2+mn8n2)+(3m2+2mn)

Answer

10m2+3mn8n2

56. (2r23rs2s2)+(5r23rs)

In the following exercises, add or subtract the polynomials.

57. (a2b2)(a2+3ab4b2)

Answer

3ab+3b2

58. (m2+2n2)(m28mnn2)

59. (p33p2q)+(2pq2+4q3)(3p2q+pq2)

Answer

p36p2q+pq2+4q3

60. (a32a2b)+(ab2+b3)(3a2b+4ab2)

61. (x3x2y)(4xy2y3)+(3x2yxy2)

Answer

x3+2x2y5xy2+y3

62. (x32x2y)(xy23y3)(x2y4xy2)

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

63. For the function f(x)=8x23x+2, find:
f(5)f(2)f(0)

Answer

187402

64. For the function f(x)=5x2x7, find:
f(4)f(1)f(0)

65. For the function g(x)=436x, find:
g(3)g(0)g(1)

Answer

104440

66. For the function g(x)=1636x2, find:
g(1)g(0)g(2)

In the following exercises, find the height for each polynomial function.

67. A painter drops a brush from a platform 75 feet high. The polynomial function h(t)=16t2+75 gives the height of the brush t seconds after it was dropped. Find the height after t=2 seconds.

Answer

The height is 11 feet.

68. A girl drops a ball off the cliff into the ocean. The polynomial h(t)=16t2+200 gives the height of a ball t seconds after it is dropped. Find the height after t=3 seconds.

69. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial function R(p)=4p2+420p. Find the revenue received when p=60 dollars.

Answer

The revenue is $10,800.

70. A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial R(p)=4p2+420p. Find the revenue received when p=90 dollars.

71. The polynomial C(x)=6x2+90x gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with x=4 feet.

Answer

The cost is $456.

72. The polynomial C(x)=6x2+90x gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 4 feet. Find the cost of producing a box with x=6 feet.

Add and Subtract Polynomial Functions

In each example, find ⓐ (f+g)(x)(f+g)(2)(fg)(x)(fg)(3).

73. f(x)=2x24x+1 and g(x)=5x2+8x+3

Answer

(f+g)(x)=7x2+4x+4
(f+g)(2)=40
(fg)(x)=3x212x2
(fg)(3)=7

74. f(x)=4x27x+3 and g(x)=4x2+2x1

75. f(x)=3x3x22x+3 and g(x)=3x37x

Answer

(f+g)(x)=6x3x29x+3
(f+g)(2)=29
(fg)(x)=x2+5x+3
(fg)(3)=21

76. f(x)=5x3x2+3x+4 and g(x)=8x31

Writing Exercises

77. Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Answer

Answers will vary.

78. Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

79. Ariana thinks the sum 6y2+5y4 is 11y6. What is wrong with her reasoning?

Answer

Answers will vary.

80. Is every trinomial a second-degree polynomial? If not, give an example.

Self Check

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

The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “identify polynomials, monomials, binomials, and trinomials”, “determine the degree of polynomials”, “add and subtract monomials”, “add and subtract polynomials”, and “evaluate a polynomial for a given value”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives 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 because topics you do not master become potholes in your road to success. In math 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 a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


This page titled 5.2E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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