Exercises: Add and Subtract Polynominals (5.1)

Exercises: Identify Polynomials, Monomials, Binomials, and Trinomials

Instructions: For questions 1-4, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

1.

a. [latex]81{b}^{5}-24{b}^{3}+1[/latex]
b. [latex]5{c}^{3}+11{c}^{2}-c-8[/latex]
c. [latex]\frac{14}{15}y+\frac{1}{7}[/latex]
d. [latex]5[/latex]
e. [latex]4y+17[/latex]

Solution

a. trinomial
b. polynomial
c. binomial
d. monomial
e. binomial

2.

a. [latex]{x}^{2}-{y}^{2}[/latex]
b. [latex]-13{c}^{4}[/latex]
c. [latex]{x}^{2}+5x-7[/latex]
d. [latex]{x}^{2}{y}^{2}-2xy+8[/latex]
e. [latex]19[/latex]

3.

a. [latex]8-3x[/latex]
b. [latex]{z}^{2}-5z-6[/latex]
c. [latex]{y}^{3}-8{y}^{2}+2y-16[/latex]
d. [latex]81{b}^{5}-24{b}^{3}+1[/latex]
e. [latex]-18[/latex]

Solution

a. binomial
b. trinomial
c. polynomial
d. trinomial
e. monomial

4.

a. [latex]11{y}^{2}[/latex]
b. [latex]-73[/latex]
c. [latex]6{x}^{2}-3xy+4x-2y+{y}^{2}[/latex]
d. [latex]4y+17[/latex]
e. [latex]5{c}^{3}+11{c}^{2}-c-8[/latex]

Exercises: Determine the Degree of Polynomials

Instructions: For questions 5-8, determine the degree of each polynomial.

5.

a. [latex]6{a}^{2}+12a+14[/latex]
b. [latex]18x{y}^{2}z[/latex]
c. [latex]5x+2[/latex]
d. [latex]{y}^{3}-8{y}^{2}+2y-16[/latex]
e. [latex]-24[/latex]

Solution

a. [latex]2[/latex]
b. [latex]4[/latex]
c. [latex]1[/latex]
d. [latex]3[/latex]
e. [latex]0[/latex]

6.

a. [latex]9{y}^{3}-10{y}^{2}+2y-6[/latex]
b. [latex]-12{p}^{4}[/latex]
c. [latex]{a}^{2}+9a+18[/latex]
d. [latex]20{x}^{2}{y}^{2}-10{a}^{2}{b}^{2}+30[/latex]
e. [latex]17[/latex]

7.

a. [latex]14-29x[/latex]
b. [latex]{z}^{2}-5z-6[/latex]
c. [latex]{y}^{3}-8{y}^{2}+2y-16[/latex]
d. [latex]23a{b}^{2}-14[/latex]
e. [latex]-3[/latex]

Solution

a. [latex]1[/latex]
b. [latex]2[/latex]
c. [latex]3[/latex]
d. [latex]3[/latex]
e. [latex]0[/latex]

8.

a. [latex]62{y}^{2}[/latex]
b. [latex]15[/latex]
c. [latex]6{x}^{2}-3xy+4x-2y+{y}^{2}[/latex]
d. [latex]10-9x[/latex]
e. [latex]{m}^{4}+4{m}^{3}+6{m}^{2}+4m+1[/latex]

Exercises: Add and Subtract Monomials

Instructions: For questions 9-32, add or subtract the monomials.

9. [latex]{7x}^{2}+5{x}^{2}[/latex]
Solution

[latex]12{x}^{2}[/latex]

10. [latex]{4y}^{3}+6{y}^{3}[/latex]
11. [latex]-12w+18w[/latex]
Solution

[latex]6w[/latex]

12. [latex]-3m+9m[/latex]
13. [latex]4a-9a[/latex]
Solution

[latex]-5a[/latex]

14. [latex]-y-5y[/latex]
15. [latex]28x-\left(-12x\right)[/latex]
Solution

[latex]40x[/latex]

16. [latex]13z-\left(-4z\right)[/latex]
17. [latex]-5b-17b[/latex]
Solution

[latex]-22b[/latex]

18. [latex]-10x-35x[/latex]
19. [latex]12a+5b-22a[/latex]
Solution

[latex]-10a+5b[/latex]

20. [latex]14x-3y-13x[/latex]
21. [latex]2{a}^{2}+{b}^{2}-6{a}^{2}[/latex]
Solution

[latex]-4{a}^{2}+{b}^{2}[/latex]

22. [latex]5{u}^{2}+4{v}^{2}-6{u}^{2}[/latex]
23. [latex]x{y}^{2}-5x-5{y}^{2}[/latex]
Solution

[latex]x{y}^{2}-5x-5{y}^{2}[/latex]

24. [latex]p{q}^{2}-4p-3{q}^{2}[/latex]
25. [latex]{a}^{2}b-4a-5a{b}^{2}[/latex]
Solution

[latex]{a}^{2}b-4a-5a{b}^{2}[/latex]

26. [latex]{x}^{2}y-3x+7x{y}^{2}[/latex]
27. [latex]12a+8b[/latex]
Solution

[latex]12a+8b[/latex]

28. [latex]19y+5z[/latex]
29. Add: [latex]4a,-3b,-8a[/latex]
Solution

[latex]-4a-3b[/latex]

30. Add: [latex]4x,3y,-3x[/latex]
31. Subtract [latex]5{x}^{6}\text{ from }-12{x}^{6}[/latex].
Solution

[latex]-17{x}^{6}[/latex]

32. Subtract [latex]2{p}^{4}\text{ from }-7{p}^{4}[/latex].

Exercises: Add and Subtract Polynomials

Instructions: For questions 33-70, add or subtract the polynomials.

33. [latex]\left(5{y}^{2}+12y+4\right)+\left(6{y}^{2}-8y+7\right)[/latex]
Solution

[latex]11{y}^{2}+4y+11[/latex]

34. [latex]\left(4{y}^{2}+10y+3\right)+\left(8{y}^{2}-6y+5\right)[/latex]
35. [latex]\left({x}^{2}+6x+8\right)+\left(-4{x}^{2}+11x-9\right)[/latex]
Solution

[latex]-3{x}^{2}+17x-1[/latex]

36. [latex]\left({y}^{2}+9y+4\right)+\left(-2{y}^{2}-5y-1\right)[/latex]
37. [latex]\left(8{x}^{2}-5x+2\right)+\left(3{x}^{2}+3\right)[/latex]
Solution

[latex]11{x}^{2}-5x+5[/latex]

38. [latex]\left(7{x}^{2}-9x+2\right)+\left(6{x}^{2}-4\right)[/latex]
39. [latex]\left(5{a}^{2}+8\right)+\left({a}^{2}-4a-9\right)[/latex]
Solution

[latex]6{a}^{2}-4a-1[/latex]

40. [latex]\left({p}^{2}-6p-18\right)+\left(2{p}^{2}+11\right)[/latex]
41. [latex]\left(4{m}^{2}-6m-3\right)-\left(2{m}^{2}+m-7\right)[/latex]
Solution

[latex]2{m}^{2}-7m+4[/latex]

42. [latex]\left(3{b}^{2}-4b+1\right)-\left(5{b}^{2}-b-2\right)[/latex]
43. [latex]\left({a}^{2}+8a+5\right)-\left({a}^{2}-3a+2\right)[/latex]
Solution

[latex]5a+3[/latex]

44. [latex]\left({b}^{2}-7b+5\right)-\left({b}^{2}-2b+9\right)[/latex]
45. [latex]\left(12{s}^{2}-15s\right)-\left(s-9\right)[/latex]
Solution

[latex]12{s}^{2}-16s+9[/latex]

46. [latex]\left(10{r}^{2}-20r\right)-\left(r-8\right)[/latex]
47. Subtract [latex]\left(9{x}^{2}+2\right)[/latex] from [latex]\left(12{x}^{2}-x+6\right)[/latex].
Solution

[latex]3{x}^{2}-x+4[/latex]

48. Subtract [latex]\left(5{y}^{2}-y+12\right)[/latex] from [latex]\left(10{y}^{2}-8y-20\right)[/latex].
49. Subtract [latex]\left(7{w}^{2}-4w+2\right)[/latex] from [latex]\left(8{w}^{2}-w+6\right)[/latex].
Solution

[latex]{w}^{2}+3w+4[/latex]

50. Subtract [latex]\left(5{x}^{2}-x+12\right)[/latex] from [latex]\left(9{x}^{2}-6x-20\right)[/latex].
51. Find the sum of [latex]\left(2{p}^{3}-8\right)[/latex] and [latex]\left({p}^{2}+9p+18\right)[/latex].
Solution

[latex]2{p}^{3}+{p}^{2}+9p+10[/latex]

52. Find the sum of [latex]\left({q}^{2}+4q+13\right)[/latex] and [latex]\left(7{q}^{3}-3\right)[/latex].
53. Find the sum of [latex]\left(8{a}^{3}-8a\right)[/latex] and [latex]\left({a}^{2}+6a+12\right)[/latex].
Solution

[latex]8{a}^{3}+{a}^{2}-2a+12[/latex]

54. Find the sum of [latex]\left({b}^{2}+5b+13\right)[/latex] and [latex]\left(4{b}^{3}-6\right)[/latex].
55. Find the difference of [latex]\left({w}^{2}+w-42\right)[/latex] and [latex]\left({w}^{2}-10w+24\right)[/latex].
Solution

[latex]11w-66[/latex]

56. Find the difference of [latex]\left({z}^{2}-3z-18\right)[/latex] and [latex]\left({z}^{2}+5z-20\right)[/latex].
57. Find the difference of [latex]\left({c}^{2}+4c-33\right)[/latex] and [latex]\left({c}^{2}-8c+12\right)[/latex].
Solution

[latex]12c-45[/latex]

58. Find the difference of [latex]\left({t}^{2}-5t-15\right)[/latex] and [latex]\left({t}^{2}+4t-17\right)[/latex].
59. [latex]\left(7{x}^{2}-2xy+6{y}^{2}\right)+\left(3{x}^{2}-5xy\right)[/latex]
Solution

[latex]10{x}^{2}-7xy+6{y}^{2}[/latex]

60. [latex]\left(-5{x}^{2}-4xy-3{y}^{2}\right)+\left(2{x}^{2}-7xy\right)[/latex]
61. [latex]\left(7{m}^{2}+mn-8{n}^{2}\right)+\left(3{m}^{2}+2mn\right)[/latex]
Solution

[latex]10{m}^{2}+3mn-8{n}^{2}[/latex]

62. [latex]\left(2{r}^{2}-3rs-2{s}^{2}\right)+\left(5{r}^{2}-3rs\right)[/latex]
63. [latex]\left({a}^{2}-{b}^{2}\right)-\left({a}^{2}+3ab-4{b}^{2}\right)[/latex]
Solution

[latex]-3ab+3{b}^{2}[/latex]

64. [latex]\left({m}^{2}+2{n}^{2}\right)-\left({m}^{2}-8mn-{n}^{2}\right)[/latex]
65. [latex]\left({u}^{2}-{v}^{2}\right)-\left({u}^{2}-4uv-3{v}^{2}\right)[/latex]
Solution

[latex]4uv+2{v}^{2}[/latex]

66. [latex]\left({j}^{2}-{k}^{2}\right)-\left({j}^{2}-8jk-5{k}^{2}\right)[/latex]
67. [latex]\left({p}^{3}-3{p}^{2}q\right)+\left(2p{q}^{2}+4{q}^{3}\right)[/latex][latex]-\left(3{p}^{2}q+p{q}^{2}\right)[/latex]
Solution

[latex]{p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}[/latex]

68. [latex]\left({a}^{3}-2{a}^{2}b\right)+\left(a{b}^{2}+{b}^{3}\right)[/latex][latex]-\left(3{a}^{2}b+4a{b}^{2}\right)[/latex]
69. [latex]\left({x}^{3}-{x}^{2}y\right)-\left(4x{y}^{2}-{y}^{3}\right)[/latex][latex]+\left(3{x}^{2}y-x{y}^{2}\right)[/latex]
Solution

[latex]{x}^{3}+2{x}^{2}y-5x{y}^{2}+{y}^{3}[/latex]

70. [latex]\left({x}^{3}-2{x}^{2}y\right)-\left(x{y}^{2}-3{y}^{3}\right)[/latex][latex]-\left({x}^{2}y-4x{y}^{2}\right)[/latex]

Exercises: Evaluate a Polynomial for a Given Value

Instructions: For questions 71-78, evaluate each polynomial for the given value.

71. Evaluate [latex]8{y}^{2}-3y+2[/latex] when:

a. [latex]y=5[/latex]
b. [latex]y=-2[/latex]
c. [latex]y=0[/latex]

Solution

a. [latex]187[/latex]
b. [latex]46[/latex]
c. [latex]2[/latex]

72. Evaluate [latex]5{y}^{2}-y-7[/latex] when:

a. [latex]y=-4[/latex]
b. [latex]y=1[/latex]
c. [latex]y=0[/latex]

73. Evaluate [latex]4-36x[/latex] when:

a. [latex]x=3[/latex]
b. [latex]x=0[/latex]
c. [latex]x=-1[/latex]

Solution

a. [latex]-104[/latex]
b. [latex]4[/latex]
c. [latex]40[/latex]

74. Evaluate [latex]16-36{x}^{2}[/latex] when:

a. [latex]x=-1[/latex]
b. [latex]x=0[/latex]
c. [latex]x=2[/latex]

75. A painter drops a brush from a platform [latex]75[/latex] feet high. The polynomial [latex]-16{t}^{2}+75[/latex] gives the height of the brush [latex]t[/latex] seconds after it was dropped. Find the height after [latex]t=2[/latex] seconds.
Solution

[latex]11[/latex]

76. A girl drops a ball off a cliff into the ocean. The polynomial [latex]-16{t}^{2}+250[/latex] gives the height of a ball [latex]t[/latex] seconds after it is dropped from a [latex]250[/latex]-foot tall cliff. Find the height after [latex]t=2[/latex] seconds.
77. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of [latex]p[/latex] dollars each is given by the polynomial [latex]-4{p}^{2}+420p[/latex]. Find the revenue received when [latex]p=60[/latex] dollars.
Solution

[latex]$10\text{,}800[/latex]

78. A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of [latex]p[/latex] dollars each is given by the polynomial [latex]-4{p}^{2}+420p[/latex]. Find the revenue received when [latex]p=90[/latex] dollars.

Exercises: Everyday Math

Instructions: For questions 79-83, answer the given everyday math word problems.
79. Fuel Efficiency. The fuel efficiency (in miles per gallon) of a car going at a speed of [latex]x[/latex] miles per hour is given by the polynomial [latex]-\frac{1}{150}{x}^{2}+\frac{1}{3}x[/latex]. Find the fuel efficiency when [latex]x=30\text{ mph}[/latex].
Solution

[latex]4[/latex]

80. Stopping Distance. The number of feet it takes for a car traveling at [latex]x[/latex] miles per hour to stop on dry, level concrete is given by the polynomial [latex]0.06{x}^{2}+1.1x[/latex]. Find the stopping distance when [latex]x=40\text{ mph}[/latex].
81. Rental Cost. The cost to rent a rug cleaner for [latex]d[/latex] days is given by the polynomial [latex]5.50d+25[/latex]. Find the cost to rent the cleaner for 6 days.
Solution

[latex]$58[/latex]

82. Height of Projectile. The height (in feet) of an object projected upward is given by the polynomial [latex]-16{t}^{2}+60t+90[/latex] where [latex]t[/latex] represents time in seconds. Find the height after [latex]t=2.5[/latex] seconds.
83. Temperature Conversion. The temperature in degrees Fahrenheit is given by the polynomial [latex]\frac{9}{5}c+32[/latex] where [latex]c[/latex] represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when [latex]c=65^\circ[/latex].
Solution

[latex]149^\circ \text{ F}[/latex]

Exercises: Writing Exercises

Instructions: For questions 84-87, answer the given writing exercises.
84. Using your own words, explain the difference between a monomial, a binomial, and a trinomial.
85. Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of [latex]5[/latex].
Solution

Answers will vary.

86. Ariana thinks the sum [latex]6{y}^{2}+5{y}^{4}[/latex] is [latex]11{y}^{6}[/latex]. What is wrong with her reasoning?
87. Jonathan thinks that [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{x}[/latex] are both monomials. What is wrong with his reasoning?
Solution

Answers will vary.

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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