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Mathematics LibreTexts

7.2E: 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. ⓐ 47x^5−17x^2y^3+y^2
5c^3+11c^2−c−8
59ab+13b
4
4pq+17

Answer

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

2. ⓐ x^2−y^2
−13c^4
a^2+2ab−7b^2
4x^2y^2−3xy+8
19

3. ⓐ 8y−5x
y^2−5yz−6z^2
y^3−8y^2+2y−16
81ab^4−24a^2b^2+3b
−18

Answer

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

4. ⓐ 11y^2
−73
6x^2−3xy+4x−2y+y^2
4y^2+17z^2
5c^3+11c^2−c−8

5. ⓐ 5a^2+12ab−7b^2
18xy^2z
5x+2
y^3−8y^2+2y−16
−24

Answer

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

6. ⓐ 9y^3−10y^2+2y−6
−12p^3q
a^2+9ab+18b^2
20x^2y^2−10a^2b^2+30
17

7. ⓐ 14s−29t
z^2−5z−6
y^3−8y^2z+2yz^2−16z^3
23ab^2−14
−3

Answer

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

8. ⓐ 15xy
15
6x^2−3xy+4x−2y+y^2
10p−9q
m^4+4m^3+6m^2+4m+1

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

9. ⓐ 7x^2+5x^2
4a−9a

Answer

12x^2−5a

10. ⓐ 4y^3+6y^3
−y−5y

11. ⓐ −12w+18w
7x^2y−(−12x^2y)

Answer

6w
19x^2y

12. ⓐ −3m+9m
15yz^2−(−8yz^2)

13. 7x^2+5x^2+4a−9a

Answer

12x^2−5a

14. 4y^3+6y^3−y−5y

15. −12w+18w+7x^2y−(−12x^2y)

Answer

6w+19x^2y

16. −3m+9m+15yz^2−(−8yz^2)

17. ⓐ −5b−17b
3xy−(−8xy)+5xy

Answer

−22b
16xy

18. ⓐ −10x−35x
17mn^2−(−9mn^2)+3mn^2

19. ⓐ 12a+5b−22a
pq^2−4p−3q^2

Answer

−10a+5b
pq^2−4p−3q^2

20. ⓐ 14x−3y−13x
a^2b−4a−5ab^2

21. ⓐ 2a^2+b^2−6a^2
x^2y−3x+7xy^2

Answer

−4a^2+b^2
x^2y−3x+7xy^2

22. ⓐ 5u^2+4v^2−6u^2
12a+8b

23. ⓐ xy^2−5x−5y^2
19y+5z

Answer

xy^2−5x−5y^2
19y+5z

24. 12a+5b−22a+pq^2−4p−3q^2

25. 14x−3y−13x+a^2b−4a−5ab^2

Answer

x−3y+a^2b−4a−5ab^2

26. 2a^2+b^2−6a^2+x^2y−3x+7xy^2

27. 5u^2+4v^2−6u^2+12a+8b

Answer

−u^2+4v^2+12a+8b

28. xy^2−5x−5y^2+19y+5z

29. Add: 4a,−3b,−8a

Answer

−4a−3b

30. Add: 4x,3y,−3x

31. Subtract 5x^6 from −12x^6

Answer

−7x^6

32. Subtract 2p^4 from −7p^4

In the following exercises, add the polynomials.

33. (5y^2+12y+4)+(6y^2−8y+7)

Answer

11y^2+4y+11

34. (4y^2+10y+3)+(8y^2−6y+5)

35. (x^2+6x+8)+(−4x^2+11x−9)

Answer

−3x^2+17x−1

36. (y^2+9y+4)+(−2y^2−5y−1)

37. (8x^2−5x+2)+(3x^2+3)

Answer

11x^2−5x+5

38. (7x^2−9x+2)+(6x^2−4)

39. (5a^2+8)+(a^2−4a−9)

Answer

6a^2−4a−1

40. (p^2−6p−18)+(2p^2+11)

In the following exercises, subtract the polynomials.

41. (4m^2−6m−3)−(2m^2+m−7)

Answer

2m^2−7m+4

42. (3b^2−4b+1)−(5b^2−b−2)

43. (a^2+8a+5)−(a^2−3a+2)

Answer

11a+3

44. (b^2−7b+5)−(b^2−2b+9)

45. (12s^2−15s)−(s−9)

Answer

12s^2−14s+9

46. (10r^2−20r)−(r−8)

In the following exercises, subtract the polynomials.

47. Subtract (9x^2+2) from (12x^2−x+6)

Answer

3x^2−x+4

48. Subtract (5y^2−y+12) from (10y^2−8y−20)

49. Subtract (7w^2−4w+2) from (8w^2−w+6)

Answer

w^2+3w+4

50. Subtract (5x^2−x+12) from (9x^2−6x−20)

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

51. Find the difference of (w^2+w−42) and (w^2−10w+24)

Answer

11w−64

52. Find the difference of (z^2−3z−18) and (z^2+5z−20)

In the following exercises, add the polynomials.

53. (7x^2−2xy+6y^2)+(3x^2−5xy)

Answer

10x^2−7xy+6y^2

54. (−5x^2−4xy−3y^2)+(2x^2−7xy)

55. (7m^2+mn−8n^2)+(3m^2+2mn)

Answer

10m^2+3mn−8n^2

56. (2r^2−3rs−2s^2)+(5r^2−3rs)

In the following exercises, add or subtract the polynomials.

57. (a^2−b^2)−(a^2+3ab−4b^2)

Answer

−3ab+3b^2

58. (m^2+2n^2)−(m^2−8mn−n^2)

59. (p^3−3p^2q)+(2pq^2+4q^3)−(3p^2q+pq^2)

Answer

p^3−6p^2q+pq^2+4q^3

60. (a^3−2a^2b)+(ab^2+b^3)−(3a^2b+4ab^2)

61. (x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)

Answer

x^3+2x^2y−5xy^2+y^3

62. (x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)

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)=8x^2−3x+2, find:
f(5)f(−2)f(0)

Answer

187402

64. For the function f(x)=5x^2−x−7, find:
f(−4)f(1)f(0)

65. For the function g(x)=4−36x, find:
g(3)g(0)g(−1)

Answer

−104440

66. For the function g(x)=16−36x^2, 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)=−16t^2+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)=−16t^2+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)=−4p^2+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)=−4p^2+420p. Find the revenue received when p=90 dollars.

71. The polynomial C(x)=6x^2+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)=6x^2+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)(f-g)(x)(f-g)(3).

73. f(x)=2x^2−4x+1 and g(x)=5x^2+8x+3

Answer

(f+g)(x)=7x^2+4x+4
(f+g)(2)=40
(f−g)(x)=−3x^2−12x−2
(f−g)(−3)=7

74. f(x)=4x^2−7x+3 and g(x)=4x^2+2x−1

75. f(x)=3x^3−x^2−2x+3 and g(x)=3x^3−7x

Answer

(f+g)(x)=6x^3−x^2−9x+3
(f+g)(2)=29
(f−g)(x)=−x^2+5x+3
(f−g)(−3)=−21

76. f(x)=5x^3−x^2+3x+4 and g(x)=8x^3−1

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 6y^2+5y^4 is 11y^6. 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 7.2E: 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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