Part I: Simplifying Algebraic Expressions (Be sure to show all steps!)
Example: -(3x + 2y – 8) = -3x -2y +8
1. –(x2 –2x+5)=________________
6. 6x-2(3x–6)+8 =___________________
2. –(3x+7)+2=_________________
7. [7(x–3)]–[6(3x-2)+x]=______________
3. -4y–(3x–7y)=________________
8. -8[2(3x+4)-2x]=__________________
4. 5y–(4x+7y)-2x=______________
9. 3{[6(x-4)+2]-[3x+2]}=_______________
5. 3(4x-7)+5=__________________
10. 2{[3(x-6)+1]-4[3(x-2)-3]}=____________
Part II: Adding and subtracting polynomials
Example: (-3x2+m) + (4m2+6m) = m2+7m
11. (12y2+17y-4) + (9y2-13y+3)=______________________
12. (7x3+4x-1) + (2x2-6x+2)=__________________________
13. (5x2-2x-1) + (3x2-5x+7)=__________________________
14. (6a3-4a2+a-9) – (3a2-5a+7)=_______________________
15. (-s2-3) – (2s2+10s)=______________________________
16. (5-9y3) + (2y2+6y-3)=_____________________________
17. (3x2-x) + 5x3 + (-4x3+x2-8)=_________________________
18. Subtract t4-3t2+7 from 5t3-9 _______________________
19. Subtract y5-y4 from y2+3y4 _________________________
20. Add 4(m2+2) to 3m2+7m ___________________________
Part III: Factoring Polynomials
Example:
a. 4x3+12x2-8x = 4x(x2+3x-2)
b. 5x3-3x2+20x-12 = x2(5x-3)+4(5x-3) = (5x-3)(x2+4)
c. x2+2x-35 = (x+7)(x-5)
d. 3x2-5x-2 = (3x+1)(x-2)
e. x2-18x+81 = (x-9)2
f. 4x2-25y2 = (2x+5y)(2x-5y)
21. x2-6x-16=________________________
26. 5x2-5=___________________________
22. 9x2-16=__________________________
27. y5+3y3+4y2+12=__________________
23. 6x2+12x+6=_______________________
28. x4-81=___________________________
24. x3+2x2-5x-10=____________________
29. 4x2-24x+36=______________________
25. 12x5-6x3+3x2=____________________
30. 25y2-4=__________________________
Part IV: Translate the following phrases into mathematical expressions (work for this section does not
need to be on a separate page)
31. The sum of ten and a number:___________________________________
32. Eighteen more than a number:___________________________________
33. Five less than a number:________________________________________
34. The product of a number and three:______________________________
35. The difference of seven and a number:____________________________
36. The difference of a number and seven:____________________________
37. Two more than a number:______________________________________
38. Sixteen less than twice a number:________________________________
39. Five times the sum of a number and four:__________________________
40. Three times the difference of a number and one:____________________
41. The quotient of a number and six:________________________________
42. Two-thirds a number:__________________________________________
43. Eight more than twice a number:_________________________________
44. Three more than the sum of a number and four:_____________________
45. The difference of a number and eight divided by ten:_________________
46. Double the difference of a number and seven:_______________________
47. Nine less than the product of a number and two:_____________________
48. The quotient of two and three more than a number:__________________
49. The product of triple a number and five:____________________________
50. Sixteen less than the sum of three and a number:_____________________
Part V: Solving equations with one unknown (Be sure to show all work on the separate page)
51. 37-x=98 _________________
61. 5a-4=26 __________________
52. -53+y=141 _______________
62. -10x-41=69 ________________
53. 59+a=-123 _______________
63. 3a+4a-3=11 ________________
54. -72+t=-40 ________________
64. 6x+5-2x=-19 _______________
55. -55=x2+32 ________________
65. 9r+3r-5=25 ________________
56. -15y=96 _________________
66. 3x+2=2x-6 _________________
57. -31t2=-93 _________________
67. 5z-4=4z-3 __________________
௫
58. ସ = −45 __________________
68. 4y+2y-7=3y+11 ______________
59. 3x+5x=48 _________________
69. 3t-5=7t+t-15 ________________
60. 18x-12x=-96 _______________
70. 6x+5x-4=2x-8 ________________
Part VI: Solving the following formulas for the given variable
Example: Solve for a:
P=2a+3b+4c
P-3b-4c=2a
௉ିଷ௕ିସ௖
ଶ
=ࢇ
71. p=mv ; for m______________________
ௗ
76. E=(0.5)kx2 ; for x___________________
ி௫
௧
72. ‫ = ݒ‬௧ ; for t_______________________
77. ܲ =
73. xf=xi+vit+(0.5)at2 ; for a______________
78. vf=vi+at ; for vi_____________________
74. Ft=m(vf-vi) ; for vi__________________
79. vf2=vi2+2ax ; for vi__________________
75. W=fx(cos(θ)) ; for x_________________
80. W=fx(cos(θ)) ; for θ_________________
; for F_____________________
Part VII: Pythagorean Theorem: Using the formula ࢇ૛ + ࢈૛ = ࢉ૛ solve for the unknown triangle side
81. a=3, b=4, c=____________
86. a=5, c=13, b=___________
82. a=6, c=10, b=___________
87. b=0.8, c=10, a=_________
83. b=12, c=13, a=__________
88. a=11, b=4, c=___________
84. a=6, c=12, b=___________
89. a=2.3, b=8.9, c=_________
85. a=8, b=6, c=____________
90. c=14, b=5.3, a=__________
Part VIII: Using the formulas for a right triangle solve for the given variable:
Formulas:
Sin ߠ =
௔
௖
Cos ߠ =
௕
௖
௔
Tan ߠ = ௕
91. a=5, c=8, sinθ=___________
94. c=2, θ=5, a=_____________
92. a=9, θ=20, b=____________
95. θ=50, b=6, a=____________
93. b=18, c=30, cosθ=________
96. a=19, b=10, tanθ=________
Part IX: solving for angles
97. c=20, b=15, θ=___________
99. b=23, c=7, θ=____________
98. a=5, c=10, θ=____________
100. a=7, b=6, θ=_____________
Part X: Substitution: Solve one equation for y in terms of x and then substitute the result to solve for x
Example:
4x+y=18
x-2y=-9
y=18-4x
x-2(18-4x)=-9
x-36+8x=-9
9x=27
x=3
101. x-y=4
106. 16(x-y)=15x+30
3x+y=36
-x+4y=5y
x=______
x=______
102. 2x-y=14
107. 8x-y=0
-x+y=-11
20x-5y+10=3(-3x-y)+13
x=______
x=______
103. 2x+y=0
108. 2(x-3)+4(y-2)=4
-4x-3y+-12
3(x-10y+2)=6(x-2y)
x=______
x=______
104. 3(x+y)=69
109. xy+3x2=168
-x-3y=-49
y+5x=-2x+17y+1
x=______
x=______
105. –x+4y=-12+x
110. 10x-y=(2x+5)(y-28)
2(3x-10y)=-78
2y+3x=3(y-x)
x=______
x=_____
Part XI: Rewrite the following numbers in scientific notation with 2 significant figures
111. 938200000=__________
116. 29799523=___________
112. 892=________________
117. 0.000295=___________
113. 0.000283=___________
118. 8987887=____________
114. 0.7536=_____________
119. 99999999=___________
115. 1050000=____________
120. 89=_________________
Part XII: Converting SI units
121. (1.5 cm) = _________m
126. (7 Mg) = _________kg
122. (6.9 km) = _________m
127. (23.4 mL)=________L
123. (800 mm) =________m
128. (90.0 g) = ________cg
124. (92 m) = _________µm
129. (5x104 pm) =______nm
125. (28 ns) = _________ms
130. (2.3x10-9s)=_______ns
Part XIII: Solving one dimensional motion problems (For these problems be sure to show all of your
work)
The following equations are used to solve problems in one dimensional motion analysis. In this class you
will be solving many word problems. Isolate the given variables and using these equations solve for the
unknowns.
Variables:
xi – initial position
vi – initial velocity
a – acceleration
Δx – change in position
Equations:
‫ݒ‬ҧ =
∆௫
௧
xf – final position
vf – final velocity
t – time
‫ݒ‬ҧ - average velocity
‫ݒ‬௙ = ‫ݒ‬௜ + ܽ‫ݐ‬
ଵ
∆‫ݒ = ݔ‬௜ ‫ ݐ‬+ ଶ ܽ‫ ݐ‬ଶ
‫ݒ‬௙ଶ = ‫ݒ‬௜ଶ + 2ܽ∆‫ݔ‬
Example:
If a vehicle is traveling at 2 m/s and accelerated at a rate of 0.4 m/s2 for 10 s how far will it travel?
Known
vi = 2 m/s
a = 0.4 m/s2
t = 10 s
Want
Δx =
Equation
1
∆‫ݒ = ݔ‬௜ ‫ ݐ‬+ ܽ‫ ݐ‬ଶ
2
݉
1
݉
∆‫ = ݔ‬ቀ2 ቁ ሺ10‫ݏ‬ሻ + ቀ0.4 ଶ ቁ ሺ10‫ݏ‬ሻଶ
‫ݏ‬
2
‫ݏ‬
∆‫ = ݔ‬40݉
Problems:
131. How fast would a vehicle be traveling if it started from rest and accelerated uniformly at a rate
of 1.2 m/s2 for a total of 5 s?
132. How long would it take for a vehicle to accelerate from 5 m/s to 8 m/s if it accelerated at a
constant rate of 0.75 m/s2?
133. What would the rate of acceleration for a bus be if it slowed from 21 m/s to a stop in 20
second?
134. How far would the bus in the previous problem travel in the stated amount of time?
135. How fast would something be going if it were to be dropped from rest 10 m above the ground
and fell with free-fall acceleration of 9.8 m/s2?
136. If a boat can stop in 50 m after traveling at 30 m/s what is the rate of acceleration?
137. How long would it take for the boat in the previous problem to stop?
138. How long would it take for a ball being dropped from rest off of a roof that is 15 m tall to hit
the ground?
139. A speeder passes a police officer traveling at a constant velocity of 30 m/s. Then the police car
starting from rest accelerated at 2.44m/s2. How much time will pass until the police car is even
with the speeder?
140. A model rocket is launched straight up with an initial velocity of 50 m/s. It accelerates with a
constant acceleration of 2 m/s2 until the engine cuts out. When that happens it is in freefall and
accelerating at -9.8 m/s2. What is the maximum height that the rocket will reach?
Answers:
131. ___________________
136. ___________________
132. ___________________
137. ___________________
133. ___________________
138. ___________________
134. ___________________
139. ___________________
135. ___________________
140. ___________________