CP College Algebra
Midterm Exam Review Packet
Name: _______________________
Date: ____________ Period: _____
Section R.1
List the numbers in each set that are (a) Natural numbers, (b) Integers, (c) Rational
numbers, (d) Irrational numbers, (e) Real numbers.
ì
1. í-4.6, 2,
î
5,
8ü
ý
2þ
3
ì
ü
2. í2.45, p ,
, 9ý
8
î
þ
ìï
1 1
16 1 ü
3. C = í 0, 1, , , - 12 , 100 , , ý
2 3
2 4þ
îï
Identify the appropriate mathematical property for each expression.
( )( ) ( )( )
4. 8 3 = 3 8
5.
( )
2 3
· =1
3 2
(
6. -6 + 6 = 0
Simplify each expression.
(
)
()
(
8. é13 - 2 - 7 + 62 - 3 4 ù
ë
û
(
10. 9 - 4
)
2
)
7. 9x + 7 4 = 36x + 28
()
(
- 5 é3 7 - 12ù
ë
û
(
) (
)
9. 3 é4 4 - 7 - 2 5 - 11 ù - 52
ë
û
)
11. 43 - 6 3 - 8 + 28 ¸ 7
)
( )
12. 6 - éë 3 × 5 + 2 2 × 3 - 2 ùû
13. 10 - éë6 - 2 × 2 + 8 - 3 ùû × 2
Simplify each expression.
14.
7 5
+
9 12
15.
2 5
9 6
16.
21 18
·
24 14
17.
40 25
¸
48 64
Section R.2
Evaluate each of the following expressions.
18. 81
19.
64
25
( )
20. -7
2
Evaluate the expression if x = –4, y = -6 and z = 8.
21.
3 y - x2
22. 2 éë4x - 7y ùû + 3z
y - x
23. 4 2z + 3y - 3 2x
Determine the restrictions on x.
x 2 +5 x -10
x3 - x
24.
Simplify each expression.
25. 4 × 2
0
æ 9ö
26. ç ÷
è 2ø
-3
-2
27. -4²
Simplify each expression. Express the answer so that all exponents are positive.
Whenever an exponent is negative or 0, we assume that the base does not equal 0.
æ 4y ö
29. ç ÷
è 5x ø
x-2 y3
28.
xy 4
(
4
-6 3
31. 2x y z
)
(-2 ) x (yz )
30.
-2
3
32 xy 3 z 4
æ x7y-5z-9 ö
32. ç -3 11 -4 ÷
èx y z ø
-4
3
Express each number in scientific notation.
33. 0.000368
34. 2514.6
Express each number in standard form.
36. 4.28 ´ 10 -5
35. 3.156 ´ 103
Section R.4
Find the degree of each polynomial.
37. 24a4b3 - 27a5b
38. 9x5yz7 - 14x3y2z4
Write each sum or difference as a polynomial in standard form.
(
) (
(
) (
)
39. 9x + 7 + 4x - 5
(
) (
)
(
) (
) (
40. 2x + 9 - 7x - 8
)
41. 5x + 11 + 4x2 - 9x + 5
4
)
42. 7x - 15 + 5x + 2 - 4x - 9
2
43. (x 2 - x + 2)+(2x 2 -3x +5)
44. ( x 2 - 3x - 4) - ( x 3 - 3x 2 + x +5)
Multiply each of the following and express your answer in standard form.
(
45. 4 7a2b - 5ab + 4ab2
(
)
)
47. 2x + 9
2
(
(
)(
(
)(
)
46. 4x - 7 3x + 5
)
48. 4x + 3 2x2 - 7x + 5
)
49. (2 x - 3) x 2 + x +1
50. ( x - 2) 3
Section R.5
Factor all polynomials completely. If the polynomial cannot be factored, write prime.
51. x 4 - 81
52. 3x 2 + 6 x - x - 2
53. 9x 2 + 1
54. 9 x 2 -12 x + 4
55. 3x 2 y - 6 xy 2 + 12 xy
56. 25x 2 - 4
57. x2 + 5x - 14
58. 6x2 - 5x - 6
60. 2x3 + 18x2 + 28x
61. 8x3 - 27
59. 3x2 - 15x - 42
62. 4x2 + 12x + 9
Section R.6
Divide each polynomial using long division.
4x3 + 4x2 - 7x - 6
63.
2x + 3
64.
x4 + 2x3 - 2x2 + 6x - 15
x2 + 3
4x 4 -15x 2 + x - 4
65.
2x - 3
Divide each polynomial using synthetic division.
3x3 - 5x2 - 17x - 12
66.
x-4
2x4 - 3x3 - 4x + 10
67.
x-2
5x 4 -3x 2 + x +1
68.
x+2
Section R.7
Perform the indicated operation and simplify the result. Leave your answer in factored
form.
69.
71.
73.
75.
x 2 - 3x - 10 x 2 + 4 x - 21
70. 2
·
x + 2 x - 35 x 2 + 9 x + 14
9 x 2 + 27 x 3
7 x + 21 x 2
x2 - 9x + 14
x2 - 8x + 7
·
x2 - 5x - 36
x2 + x - 12
2x
2
x + 8x + 7
4
77.
3y
+
1
-
-
¸
x2 + 3x - 4
72.
x2 + 3x - 10
x2 - 4x - 45
74.
x2 + 2x - 15
4x
76.
2
5x - 5
3
78.
4 x 12 xy
8x
79. x - 1
10 x
2
80.
x +1
2x2 + 5x + 2 2x2 + x - 1
· 2
4x2 - 1
x +x-2
6x2 + 13x + 6
4x2 - 9
4x
2
x - 2x
+
¸
6x2 + x - 2
4x2 - 1
2
3x + 6
x
2x - 3
+
x + 1 x -1
1+
1-
1
x
1
x
Section R.8
Simplify each expression.
81.
3
-16
82.
3xy 3 2 x 2 y
6x y
3
4
83. 2 3 8 x + 2 3 64 x - 3 3 125 x
84. ( 8 + 5)( 8 - 3)
85. 96a12
87. 3 32a20
86. 54a7b12
89. 12 6 - 7 6 - 14 6
(
88. 3 81a22b10
90. 5 12 + 4 27
)( )
(
)(
92. 6 + 5 2 9 - 4 2
91. 9 24 4 8
Rationalize the denominator of each expression.
3
93.
94.
5+ 2
)
3- 2
3+ 2
Simplify each expression. Express your answer so that only positive exponents occur.
95. 9
-3
1
96. ( x 2 y ) 3 ( xy 2 )
2
2
3
( )
97. 24
2 1 ö
æ
36x 3 y 6 ÷
ç
98.
-2 ÷
ç
çè 81x -2y 3 ÷ø
3
2
Section 1.1
Solve each linear equation for x.
99. 5x -
(
1
3
= 2x 3
2
(
)
101. 7x - 6 11 - 2x = 10
103)
2x
x+3
=
-6
x+3
)
(
100. 3 x - 2 - 5 = 8 - 2 x - 4
-2
Solve each rational equation for x.
102. 4x -
)
1
4
= 3x +
10
5
104) 8 x - (3x + 2)= 3 x -10
-
1
2
105.
2
3x - 5
=
4
x - 15
106.
7x
x -4
2
+
5
x-2
=
2x
107.
x -4
2
x
x -3
-
7x - 6
x -x-6
2
=
2
x +2
Solve each literal equation for x.
108.
b-c
= m; x ¹ a
x-a
109. a - b=
d e
+ ; x¹0
x x
110.
b+c b-c
=
; c ¹ 0 and a ¹ 0
x+a x-a
Section 1.2
Use the discriminant to determine the nature of the solutions.
111. 2 x 2 - 6 x + 7 = 0
112. 4x 2 -12x +9=0
Solve each quadratic equation by factoring.
113. 24x2 + 42x = 0
114. x2 + 5x - 36 = 0
115. 8x2 + 26x + 15 = 0
Solve each quadratic equation by the square root property.
116. ( x + 2)2 = 1
117. x 2 - 25 = 5
Solve each quadratic equation by completing the square.
118. x2 + 12x + 4 = 0
119. 2x2 + x - 28 = 0
Solve each quadratic equation by using the quadratic formula.
120. 4x2 - 4x - 3 = 0
121. 7x2 - 2x - 25 = 0
Section 1.3
Write each expression in the standard form of a + bi.
3
122. -25
123.
3+ i
124. i 50
125. i -23
(
) (
(
)
)
126. 7 + 5i + 4 - 11i
129. 3 + 4i
2
(
) (
)
(
127. 9 - 7i - 16 - 5i
130.
)(
)
128. 5 - 4i 6 + 5i
4 + 2i
5 - 3i
131. 11i37 - 19i15 + 7i21
Solve each quadratic equation for x in the complex number system.
132. x2 + 49 = 0
133. x4 - 16 = 0
134. 9x2 + 3x + 4 = 0
Section 1.4
Solve each radical equation for x.
135. 3 1- 2 x - 3 = 0
137. 5 7x + 4 - 2 = 23
136.
138.
x - 2 = 2 + 2x + 3
Solve each quadratic equation for x using substitution.
139. x4 + 5x2 + 6 = 0
(
15 - 2 x = x
)
2
(
)
140. x - 1 - 4 x - 1 - 21 = 0
Section 1.5
Solve each inequality. Express your answer in interval notation. Graph the solution.
141. x + 1 < 2x – 5
142. 3x – 7 > 2
143. -3 <
2x - 1
< 0
4
144. -12 < -5x + 24 < 36
Section 1.6
Solve the equation. Express your answer in a solution set.
145. 2x + 3 = 5
146. -2 3x - 5
= -12
Solve each inequality. Express your answer in interval notation. Graph the solution.
147. x - 2 + 2 < 3
148. x - 3 ³ 2
Section 1.7
Solve each problem as indicated.
149. Trent can deliver his newspapers in 30 minutes. It takes Lois 20 minutes to do the
same route. How long would it take them to deliver the newspapers if they work
together?
150. A coffee house has 20 pounds of a coffee that sells for $4 per pound. How many
pounds of a coffee that sells for $8 per pound should be mixed with the 20 pounds of $4
per pound
coffee to obtain a blend that will sell for $5 per pound? How much of the $5 per pound
coffee is there to sell?