# 025test2reviewfall09 97 03

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```					                                                                                           MATH 025
TEST 2
REVIEW SHEET
TO THE STUDENT:

This Review Sheet gives you an outline of the topics covered on Test 2 as well as practice
problems. Answers are at the end of the Review Sheet.

I. EXPRESSIONS:

A. Simplifying Expressions

For problems 1 – 7, simplify each of the following exponential expressions.
2
2 5
1. 3x x                                                       3x7 
5.       2
20m8                                                       6x 
2
2. 4m                                                       3   2    5 2 3
4                                         6. (5 x y ) (4 x y )
 3x 
     
 x  y  x  y
5           4
3.  5                                              7.
4. (3ab)
3

For problems 8 – 21, simplify each of the following expressions. All the answers should
have all positive exponents.
1
 a                                             8 x 4 y 2
 
 bc 
16. 10 x y
3 4
8.

9.
x 
3  5
a 5
3                                        3
 2y                                      17. a
 2
10.  x                                               2 y 
3

2  3x 
2

18. 2 y
3
11.
5a 2b 4                                         1
3                                          4
12. 10a b                                         19. x
13.     n7 n2 n0
2                             20.
x y 
2   3 4

 4 x 3 a 
       2                                           a 2b 3
14.      3 xa                                      21.
a 5b 3
x 3 y 2
2 3
15. x y

B. Multiplying Polynomials
For problems 22 – 41, simplify each of the following expressions.
22.    3  5( x 2  4 x  2)                         23.     8a  3a(4  a )
24.   ab  b2  a3                                 33.    x  3 x  3
5 y  3 y  y  4                            34. 4  x  2 
2
25.
26.   2 x(3 x 2  4 x  2)                         35.  5  4 x 5  4 x 
27.    x  y   x2  xy  y 2                    36.  6x  2 x  6

 a  4                                     37.  x  8
2                                                2
28.
29.   2  4  2 x  x2   x  4  2 x  x2        38. 3x  43x  4

30.    2 x 1 x  2                             39.  4x  2 4x  2
31.    3x  12  4 x                            40.  2 x  3  4 x2  6 x  9 
32.     x 15x  3
41. Does  a  b   a 2  b2 ? Why? Explain clearly.
2

C. Factoring
For problems 42 – 65, factor each of the following expressions completely. If a
polynomial cannot be factored label it “prime”.
42. 15x 2  9 x  6                               54. x 2  8 x  12
43. 25 x 2 y  30 xy 2  10 x 2 y                55. 16 x 2  100
44. 36 x 2 y  24 xy 2                            56. x 2  x  4
57. x 2  13x  30
45. 8a3b  6a 2b  4a 2b2
58. 6 x 2  x  2
46. 18mn  6m 2
59. x 2  49
47. 5x 3x  2  2 3x  2
60. 5  6x  x 2
48. 8x  x  3   x  3
2
61. 64  y 2
49. x  x  5   x  5
2
62. 2n 2  3n  9
50. 9a 2  36a  36                               63. 6 x  150
2

51. 2a 3  18a                                    64. 5x  19 x  4
2

52. 4t  12t  9
2
65. 9 x 3  3x 2  6 x
53. 3t 2  18t  15

OLD TOPICS
For problems 66 – 69 use the distributive property and/or combining like terms to
simplify.
66. 3( x + 4)                                  68.       8 – 2(x – 1)
67. 4x2 – 3xy – 2y2 – 6x2 + 6x                   69.       3a + 3(a – 2) – (2a + 5)

2
II. EQUATION

A. Solving polynomial equations
For problems 70 – 78, solve each equation by factoring.
70. z 2  5 z  0                                 75. x 3x  2  5
71. 9n 2  4  0                                                         76. 30 y 2  25 y  5  0
72. x  x  20  0                                                       77. y 3  3 y 2  4 y  0
2

73. x  81x
4      2
78. 25 x 2  50 x  75
74. 2 x  3x  14
2

For problems 79 – 83 use your graphing calculator to solve the equations graphically.
79. x 2  25  0                                 82. 25 x 2  4
80. 2 x 2  18 x  28                                                    83. x3  3x 2  4 x
81. 9t 2  4t

84. The graph of f ( x)  x 3  3 x 2  10 x is given below. Label on the graph the solutions to the
equation x3  3x 2  10 x  0 , then state the solutions separately.
y = x^3+3x^2-10x

















                                      


 

 

 

85. a. Use a graphing calculator to complete the table below.
b. Circle the solutions to x3  x 2  6 x  0 in the table, then state the solutions separately.
x         y  x3  x 2  6 x
3
2
1
0
1
2
3
4

3
OLD TOPICS
B. Linear Equations
For problems 86 and 87, a) use a graphing calculator to complete the tables below
(show the line of the solution, and a line above and a line
below the solution, then state your solution separately).
b) solve the equation using algebra.

86.     2 x  3  5  8x                                 87. 7  2( x 1)  5( x 1)
x Y1            Y2
x    Y1           Y2

C. Systems of Linear Equations
For problems 88 – 90, solve the system of equations by using a graphing calculator.
a) Copy the graphs on a graph paper using the same axis.
b) Find the solution as an ordered pair.

x  y  2                           8 x  6 y  4                          x  y  3
                           89.                                     90.     
88.       2                                   3 y  4 x  2                          3x  3 y  9
3 x  y  3


For problems 91 and 92, solve the systems of equations by substitution or elimination.
2 x  3 y  7                                       x  2 y  11
91.                                               92.     
3 y  4 x  1                                     3x  4 y  7
x  y  2

93.   Use a graphing calculator to solve the system  2            numerically. Show three
3 x  y  3

lines of your table and state the solution separately.

D. Equations of Lines

94. A line goes through the points (–10, –1) and (10, 1). Write the equation of this line.

95. Find the equation of the line which passes through the two points (7, 1), and (2, 1).
5
96. Find the equation of the line which has slope and passes through the point (–4, 2).
2
97. Find the equation of the vertical line that passes through the point (5, –3).

E.      Linear Inequalities
98. a)           Solve 4  2 x  3(2 x  4) using algebra.
b) Express the answer in interval notation.
c) Graph the solution set on a number line.

4
99. a) Solve 4  3x  8  x using algebra.
b) Express the answer in interval notation.
c) Graph the solution set on a number line.

For problems 100 and 101 solve the systems of inequalities.

 x  y  3                                           y  2
100.                                                  101.    
2 x  y  2                                          3 y  x  3

III. APPLICATION PROBLEMS

A. Scientific Notation

102.   Change 32,794,810 kilometers to scientific notation.
103.   Change 5.39  104 kilograms from scientific notation to standard form.
104.   The mass of an average chicken is 1,800 grams. Change to scientific notation.
105.   Dinosaurs were extinct by 6.5 107 years ago. Change to standard form.

B. Polynomial Expressions

For problems 106 and 107, consider the following shapes.

3x2
2x            3x 2
3x  4

106. Write expressions for the circumference and the area of the circle.
107. Write an expression for the volume of the rectangular box.

C. Polynomial Equations

108. The height of a golf ball in feet after t seconds is given by h  96t  16t 2 . How long did
it take for the ball to hit the ground?

109. The braking distance D in feet required to stop a car traveling x miles per hour on wet,
1
level pavement is approximated by D  x 2 .
9
a) Calculate the braking distance for 30 miles per hour and 60 miles per hour. How do
b) If the braking distance is 25feet, find the speed of the car.
agree?

5
OLD TOPICS
5
110. Solve the formula C   F  32       for F.
9
1
111. Solve the formula A      h(a  b) for h .
2

For problems 112 – 115, solve each problem using an algebraic approach. You may
wish to set up a system of two equations in two variables or you may wish to set up one
equation in one variable. Use whatever tool is appropriate for the problem. Make sure
to define each variable you use.

112. Joella told her brother that she earned \$650 in interest that year. She started the year
with \$10,000 and invested part of it in stocks at a 7.5% interest rate and she invested the
rest in bonds at a 5% interest rate. Determine how much she invested in stocks and how
much in bonds.

113. A chemist working on a flu vaccine needs to mix 10% sodium – iodine solution with a
60% sodium – iodine solution to obtain 50 milliliters of a 30% sodium – iodine solution.
How many milliliters of the 10% solution and of the 60% solution should be mixed?

114.     A vegetable garden is in the shape of a rectangle in which the length is 3 feet more
than twice the width. If the perimeter of the garden is 72 feet, what is the length and
width of the garden?

115.     A tugboat goes 120 miles upstream in 15 hours. The return trip downstream takes 10
hours. Find the speed of the tugboat without a current and the speed of the current.

For problems 116 – 118 use unit analysis to change:

116.    10 kilometers to miles. (Useful fact: 1 mile  1.6km)
117.    60 km/h to feet per second. (Useful fact: 1 mile  1.6km)
118.    3 cubic yards to cubic inches.

6
MATH 025
Test 2 Review

1.    3x7                         31.   12 x 2  10 x  2             65. 3x  x  2 x  1
2.    5m        6                 32.   5x2  2 x  3                  66. 3x  12
81x 4                       33.   x2  9                         67. 2 x 2  2 y 2  3xy  6 x
3.
625                         34.   4 x 2  16 x  16              68. 2 x  10
4.    27a3b3                     35.   25  16x 2                     69. 4a  11
x10                         36.   6 x 2  10 x  12              70. z = 0, z = 5
5.
4                          37.   x2  16 x  64                 71. n  2 , n   2
6.    1600x 21 y 8                38.   9 x2  16                                       3                               3
39.   16 x 2  16 x  4              72. x  4 , x  5
 x  y
9
7.
40.   8x2  27                       73. x  0 , x  9 , x  9
8.    bc
41.   No. Explain.                   74. x  2 , x   7
3  5 x 2  3x  2 
a
42.                                                                                    2
9.    x15
10.    x6                                        
43. 5xy 5x  6 y  2 x      2
   75. x   5 , x  1
3
8 y3                        44. 12xy  x  y                            1,      1
76. y      y
11. 18x 2
3
45. 2a b  4a  2b  3
2                                  3       2
12. b                                                                  77. x  1 , x  0 , x  4
46. 6m  3n  m
2a
78. x  1 , x  3
13. 1                             47.   3x  25x  2               79. x  5, x  5
n9
14. 16 x
4                        48.    x  3 8x2  1                                                

9a 2                                x  5  x2  1
                                                  

49.                                                                    
5
15. x
9  a  2
2                                                     

50.
y5                                                                                                   

16. 4 x                           51. 2a  a  3 a  3                                                

2                                                                                            
5y
 2t  3
2
52.                                                                    

17. 1                                                                  80. x = -10.35; x = 1.35
a8                            53. 3  t  5 t  1                                                                 





18. 4                             54.    x  6 x  2                                                                 



19. x 4                                                                                                                   

20. x 8
55.   4  2x  5 2x  5                                         


 
          

 

y12                         56. Prime                                                                              

57.  x  3 x 10
 

21.   a3                                                                                                                 

 

22.   5x2  20 x  7             58.    2x 13x  2
 

81. t  0, t  4 / 9
23.   3a 2  20a
24.   ab3  a 4b3
59.    x  7 x  7                                                        

25.   17 y  3 y 2                60.     x  5 x 1                                                       

26. 6 x  8x  4 x
3       2         61. Prime
62.  2n  3 n  3


27. x 3  y 3
28. a 2  8a  16                 63. 6  x  5 x  5                                                                                      

29. 8  x 3                       64.  5x 1 x  4                                                        

30. 2 x 2  3x  2

7
100.

82. No Solution                                                                                                    89. No Solution                                                                                            

                                                                         

                                                                                                                                                                           


                                                                                                 





                                                       

                                                                                         

                                                    



                                                                                                                                                                           




                                                                                                    




101.
83. x  4, x  0, x  1                                                                                                                                                                             

y = x^3+3x^2-4x



90. Infinitely many solutions

(coincident lines)
                                                                                            



                                                                                                                                             


                                                                                                                                                                                                  


                                             
                                                                                            
                                             

                                                              

                                                                                           


102.            3.28 107
84. x  5 , x  0 , x  2                                                                                                                         


103.        0.000539
85. a.
x                                                                                                                                                                                  104.         1.8 103
y  x3  x 2  6 x                                                                                                                    
105.          65,000,000
3                                                        0                               91. x  1, y   5                                                    106.         C  6 x 2 ,
2                                                            8                                                3
1                                                            6                               92. x  3, y  4                                                                   A  9 x 4
0                                                         0                                                                                                     107.         V  18x 4  24 x3
1                                                         4
93. Solution x  3, y  1                                             108.         6 seconds
2                                                          0                               X    Y1=x-2 Y2=-2/3x+3                                               109. a) 100 ft; 400ft
3                                                         18                               2    0         5/3                                                           b) 15mph
4                                                         56                               3    1         1                                                            c) check with calculator
b. x  3 , x  0 ,                                                                                x2          4    2         1/3                                                              9
110.   F  C  32
86. x=0.8                                                                                                          94. y  1 x                                                                      5
x                            Y1                                          Y2                           10                                                   111. h   2A
95. y  1                                                                      ab
0.7                                                   -1.6                                        -0.6
112. \$4000 in bonds; \$6000
0.8                                                   -1.4                                        -1.4        96. y  5 x  12                                                              in stocks
0.9                                                   -1.2                                        -2.2                 2
113. 30 ml of 10% soln; 20
87. x  2                                                                                                          97. x  5                                                                     ml of 60% soln.
x                                                                                    98. a. x  2                                                          114. W=11ft; L =25ft
Y1                                          Y2
b. (, 2]                                                        115.    Speed of boad is
1                                                       7                                           0                                                    ]                             10mph; speed of current is
c.
2                                                      5                                           5                                                    -2
2mph
99. a. x  1
3                                                       3                                           10                                                                            116.            1mile
b. (1, )                                                                 10km          6.25miles
88. (3, 1)                                                                                                                                                                                               1.6km
y = x-2
y = -2/3x+3

c.                                            (                   117.                 55 ft/sec
1
60km 1mile 5280 ft 1hr   1min      ft
                       55


                                                                                                                                  hr 1.6km 1mile 60 min 60sec       s

118. 139968in3
                                                                                        
36in 36in 36in
                                                                                                                                 3 yd 3                139968 in3
                                                                                                                                          1yd 1yd 1yd






8

```
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