# 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. $81{b}^{5}-24{b}^{3}+1$
b. $5{c}^{3}+11{c}^{2}-c-8$
c. $\frac{14}{15}y+\frac{1}{7}$
d. $5$
e. $4y+17$

Solution

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

2.

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

3.

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

Solution

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

4.

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

## Exercises: Determine the Degree of Polynomials

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

5.

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

Solution

a. $2$
b. $4$
c. $1$
d. $3$
e. $0$

6.

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

7.

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

Solution

a. $1$
b. $2$
c. $3$
d. $3$
e. $0$

8.

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

## Exercises: Add and Subtract Monomials

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

9. ${7x}^{2}+5{x}^{2}$
Solution

$12{x}^{2}$

10. ${4y}^{3}+6{y}^{3}$

11. $-12w+18w$
Solution

$6w$

12. $-3m+9m$

13. $4a-9a$
Solution

$-5a$

14. $-y-5y$

15. $28x-\left(-12x\right)$
Solution

$40x$

16. $13z-\left(-4z\right)$

17. $-5b-17b$
Solution

$-22b$

18. $-10x-35x$

19. $12a+5b-22a$
Solution

$-10a+5b$

20. $14x-3y-13x$

21. $2{a}^{2}+{b}^{2}-6{a}^{2}$
Solution

$-4{a}^{2}+{b}^{2}$

22. $5{u}^{2}+4{v}^{2}-6{u}^{2}$

23. $x{y}^{2}-5x-5{y}^{2}$
Solution

$x{y}^{2}-5x-5{y}^{2}$

24. $p{q}^{2}-4p-3{q}^{2}$

25. ${a}^{2}b-4a-5a{b}^{2}$
Solution

${a}^{2}b-4a-5a{b}^{2}$

26. ${x}^{2}y-3x+7x{y}^{2}$

27. $12a+8b$
Solution

$12a+8b$

28. $19y+5z$

29. Add: $4a,-3b,-8a$
Solution

$-4a-3b$

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

31. Subtract $5{x}^{6}\text{ from }-12{x}^{6}$.
Solution

$-17{x}^{6}$

32. Subtract $2{p}^{4}\text{ from }-7{p}^{4}$.

## Exercises: Add and Subtract Polynomials

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

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

$11{y}^{2}+4y+11$

34. $\left(4{y}^{2}+10y+3\right)+\left(8{y}^{2}-6y+5\right)$

35. $\left({x}^{2}+6x+8\right)+\left(-4{x}^{2}+11x-9\right)$
Solution

$-3{x}^{2}+17x-1$

36. $\left({y}^{2}+9y+4\right)+\left(-2{y}^{2}-5y-1\right)$

37. $\left(8{x}^{2}-5x+2\right)+\left(3{x}^{2}+3\right)$
Solution

$11{x}^{2}-5x+5$

38. $\left(7{x}^{2}-9x+2\right)+\left(6{x}^{2}-4\right)$

39. $\left(5{a}^{2}+8\right)+\left({a}^{2}-4a-9\right)$
Solution

$6{a}^{2}-4a-1$

40. $\left({p}^{2}-6p-18\right)+\left(2{p}^{2}+11\right)$

41. $\left(4{m}^{2}-6m-3\right)-\left(2{m}^{2}+m-7\right)$
Solution

$2{m}^{2}-7m+4$

42. $\left(3{b}^{2}-4b+1\right)-\left(5{b}^{2}-b-2\right)$

43. $\left({a}^{2}+8a+5\right)-\left({a}^{2}-3a+2\right)$
Solution

$5a+3$

44. $\left({b}^{2}-7b+5\right)-\left({b}^{2}-2b+9\right)$

45. $\left(12{s}^{2}-15s\right)-\left(s-9\right)$
Solution

$12{s}^{2}-16s+9$

46. $\left(10{r}^{2}-20r\right)-\left(r-8\right)$

47. Subtract $\left(9{x}^{2}+2\right)$ from $\left(12{x}^{2}-x+6\right)$.
Solution

$3{x}^{2}-x+4$

48. Subtract $\left(5{y}^{2}-y+12\right)$ from $\left(10{y}^{2}-8y-20\right)$.

49. Subtract $\left(7{w}^{2}-4w+2\right)$ from $\left(8{w}^{2}-w+6\right)$.
Solution

${w}^{2}+3w+4$

50. Subtract $\left(5{x}^{2}-x+12\right)$ from $\left(9{x}^{2}-6x-20\right)$.

51. Find the sum of $\left(2{p}^{3}-8\right)$ and $\left({p}^{2}+9p+18\right)$.
Solution

$2{p}^{3}+{p}^{2}+9p+10$

52. Find the sum of $\left({q}^{2}+4q+13\right)$ and $\left(7{q}^{3}-3\right)$.

53. Find the sum of $\left(8{a}^{3}-8a\right)$ and $\left({a}^{2}+6a+12\right)$.
Solution

$8{a}^{3}+{a}^{2}-2a+12$

54. Find the sum of $\left({b}^{2}+5b+13\right)$ and $\left(4{b}^{3}-6\right)$.

55. Find the difference of $\left({w}^{2}+w-42\right)$ and $\left({w}^{2}-10w+24\right)$.
Solution

$11w-66$

56. Find the difference of $\left({z}^{2}-3z-18\right)$ and $\left({z}^{2}+5z-20\right)$.

57. Find the difference of $\left({c}^{2}+4c-33\right)$ and $\left({c}^{2}-8c+12\right)$.
Solution

$12c-45$

58. Find the difference of $\left({t}^{2}-5t-15\right)$ and $\left({t}^{2}+4t-17\right)$.

59. $\left(7{x}^{2}-2xy+6{y}^{2}\right)+\left(3{x}^{2}-5xy\right)$
Solution

$10{x}^{2}-7xy+6{y}^{2}$

60. $\left(-5{x}^{2}-4xy-3{y}^{2}\right)+\left(2{x}^{2}-7xy\right)$

61. $\left(7{m}^{2}+mn-8{n}^{2}\right)+\left(3{m}^{2}+2mn\right)$
Solution

$10{m}^{2}+3mn-8{n}^{2}$

62. $\left(2{r}^{2}-3rs-2{s}^{2}\right)+\left(5{r}^{2}-3rs\right)$

63. $\left({a}^{2}-{b}^{2}\right)-\left({a}^{2}+3ab-4{b}^{2}\right)$
Solution

$-3ab+3{b}^{2}$

64. $\left({m}^{2}+2{n}^{2}\right)-\left({m}^{2}-8mn-{n}^{2}\right)$

65. $\left({u}^{2}-{v}^{2}\right)-\left({u}^{2}-4uv-3{v}^{2}\right)$
Solution

$4uv+2{v}^{2}$

66. $\left({j}^{2}-{k}^{2}\right)-\left({j}^{2}-8jk-5{k}^{2}\right)$

67. $\left({p}^{3}-3{p}^{2}q\right)+\left(2p{q}^{2}+4{q}^{3}\right)$$-\left(3{p}^{2}q+p{q}^{2}\right)$
Solution

${p}^{3}-6{p}^{2}q+p{q}^{2}+4{q}^{3}$

68. $\left({a}^{3}-2{a}^{2}b\right)+\left(a{b}^{2}+{b}^{3}\right)$$-\left(3{a}^{2}b+4a{b}^{2}\right)$

69. $\left({x}^{3}-{x}^{2}y\right)-\left(4x{y}^{2}-{y}^{3}\right)$$+\left(3{x}^{2}y-x{y}^{2}\right)$
Solution

${x}^{3}+2{x}^{2}y-5x{y}^{2}+{y}^{3}$

70. $\left({x}^{3}-2{x}^{2}y\right)-\left(x{y}^{2}-3{y}^{3}\right)$$-\left({x}^{2}y-4x{y}^{2}\right)$

## Exercises: Evaluate a Polynomial for a Given Value

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

71. Evaluate $8{y}^{2}-3y+2$ when:

a. $y=5$
b. $y=-2$
c. $y=0$

Solution

a. $187$
b. $46$
c. $2$

72. Evaluate $5{y}^{2}-y-7$ when:

a. $y=-4$
b. $y=1$
c. $y=0$

73. Evaluate $4-36x$ when:

a. $x=3$
b. $x=0$
c. $x=-1$

Solution

a. $-104$
b. $4$
c. $40$

74. Evaluate $16-36{x}^{2}$ when:

a. $x=-1$
b. $x=0$
c. $x=2$

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

$11$

76. A girl drops a ball off a cliff into the ocean. The polynomial $-16{t}^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a $250$-foot tall cliff. Find the height after $t=2$ seconds.

77. 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 $-4{p}^{2}+420p$. Find the revenue received when $p=60$ dollars.
Solution

$10\text{,}800$

78. 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 $-4{p}^{2}+420p$. Find the revenue received when $p=90$ 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 $x$ miles per hour is given by the polynomial $-\frac{1}{150}{x}^{2}+\frac{1}{3}x$. Find the fuel efficiency when $x=30\text{ mph}$.
Solution

$4$

80. Stopping Distance. The number of feet it takes for a car traveling at $x$ miles per hour to stop on dry, level concrete is given by the polynomial $0.06{x}^{2}+1.1x$. Find the stopping distance when $x=40\text{ mph}$.

81. Rental Cost. The cost to rent a rug cleaner for $d$ days is given by the polynomial $5.50d+25$. Find the cost to rent the cleaner for 6 days.
Solution

$58$

82. Height of Projectile. The height (in feet) of an object projected upward is given by the polynomial $-16{t}^{2}+60t+90$ where $t$ represents time in seconds. Find the height after $t=2.5$ seconds.

83. Temperature Conversion. The temperature in degrees Fahrenheit is given by the polynomial $\frac{9}{5}c+32$ where $c$ represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when $c=65^\circ$.
Solution

$149^\circ \text{ F}$

## 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 $5$.
Solution

86. Ariana thinks the sum $6{y}^{2}+5{y}^{4}$ is $11{y}^{6}$. What is wrong with her reasoning?

87. Jonathan thinks that $\frac{1}{3}$ and $\frac{1}{x}$ are both monomials. What is wrong with his reasoning?
Solution