College Algebra Review Add or Subtract: 1.
( 2x
2
) (
) (
)
+ 11xy − 15 + 6x 2 − 6x + 2 − −x 2 − y − 5 Multiply: 2.
( x − 2 ) ( x + 7 ) 6.
( 7m + 9 )2 3.
( 2a − 5b ) ( 5a + b ) ( 6a − 5 )2 4.
( x − 4 ) ( x + 4 ) ( x 2 + 16 ) 7.
8.
5.
(11 − 2c ) (11 + 2c ) ( −2x
9.
( 5x − 6y + 2 ) ( 5x − 6y − 2 )
2
)(
)
− 6y + 2 5x 2 + 9y + z Factor out a common factor: 10. 17x − 21x 4 13. 3a 2 − 12a + 27 11. 54y 2 − 6y 7 14. 6x ( 3x + 4 ) − 5 ( 3x + 4 )
12. 4n 2 + 12n − 32 Factor by grouping: 15. x 3 + 7x 2 + 5x + 35 3
2
16. 9a − 9a + 18a − 18
Factor the trinomial: 17. s 2 + 6s + 8 23. 20x 2 − 17xy − 10y 2 18. u 2 − 12u + 27 24. 4x 2 + 8x − 60 19. x 2 − 4x − 60 25. x 3 + 4x 2 + 3x 20. 5x 2 − 37x − 72 26. 3y 3 − 3y 2 − 18y 21. 6x 2 − 49x + 49 22. 15z 2 + 8z − 16 2
27. 4y + 18y − 10
Factor the difference of squares: 28. x 2 − 144 30. 25y 4 − 9 29. 25x 2 − 16 2
2
31. 81k y − 25ym
1 Factor: 32. x 2 + 20x + 100 36. 1 − 6x + 9x 2 33. r 2 − 10r + 25 37. 1 + 4x + 4x 2 34. z 2 + 14z + 49 38. a 3 − 14a 2 + 49a 35. 49x 2 − 28x + 4 2
2
39. 11p − 22 pq + 11q
Factor the sum or difference of cubes: 40. x 3 − 343 43. 7b 5 − 56b 2 41. t 3 + 729 3
44. 27x + 125
42. 8y 3 − 8 Factor completely: 45. 2a 2 b − 10ab 2 47. 81x 2 − 2 46. 72x 2 − 128 2 2
48. a b + 8ab + 16
Simplify: 49. 121 50.
( −3)2
51.
3
−27 Simplify: (Assume that variables can represent any real number) 52.
55. 162k 6 q 8 441t 2 53. 3 125k 12 54.
56.
3
−64x 7 y 5
3
81x 4
3x
z 2 − 12z + 36 Simplify: 57.
7 14 58.
2xy 14xy 8 59. 3 12 3 8 60.
14ab
7a
61.
62. −8 11 − 19 11 63. −5 3 − 8 75
2 Simplify: ( 12 + 4 ) ( 12 − 4 ) 65. ( 2 2 + 5 3 ) ( 6 2 + 4 3 ) 64.
(
(
67.
66. 2 + 2
7−
) 5)
2
2
Solve: 68. A long-­‐distance runner runs 5 miles south and then 8 miles east. How far is the runner from the starting point? ? 5 mi 8 mi 69. The diagram below shows a rope connecting the top of a pole to the ground. The rope is 29 yd. long and touches the ground 25 yd from the pole. How tall is the pole? 29 yd ? 25 yd Rationalize the denominator: 5
6
70.
2
71.
8− 7
3
72.
6+7
73.
2− 7
2+ 7
74.
3
7
5
75.
3
4
9
Convert to radical notation and simplify: 2
7
76. x 4
3
77. 27 78. 216
4
5
79. a b
−
2
3
−
1
5
3 Convert to exponential notation and simplify: 80.
3
81.
( 11) 36 82.
6
5
83.
3
27 2 4
815
Simplify and then, if appropriate, write in radical notation: ⎛ 67 ⎞ ⎛ 13 ⎞
84. ⎜ 5a ⎟ ⎜ 3a ⎟ ⎝
⎠⎝
⎠
5
⎛ 3 5⎞3
85. ⎜ m 2 n 2 ⎟ ⎝
⎠
Solve: 86. 3 ( 2z − 5 ) = 5 ( z + 4 ) 87. 2 ( x + 7 ) = 7 − 4 ( x + 7 ) 88. 12 − 5 ( x + 4 ) = 15 − 8 ( x + 1) 89. x = 9 90. x − 100 = 0 91. 3x 2 − 33 = 0 92. 64k 2 = 16 93. x 2 − 8x = 0 94. a 2 − 12a + 20 = 0 2
2
95. y 2 + 49 = 14y 96. 4x 2 − 24x + 32 = 0 97. x ( x − 4 ) = 45 98. 2k − 15k − 8 = 0 99. 5x 2 − 16x = 16 2
4 Add or Subtract, and, if possible, simplify. 100. 3
2a
+
2a + 3 2a + 3 102. 101. 12
5
+ 2
2
x y xy y
2
−
y − y − 20 y + 4 2
Simplify. 103. 104. x 2 + 2x − 3
x 2 − 9 Simplify the next two problems without using a calculator. Show a−b
b
a 2 − b2
ab all your work to get credit. 2
105. 3⋅ 2 − 4 ⋅ 2 + 6(3− 1) 4(8 − 6)2 − 4 ⋅3+ 2 ⋅8
31 + 190
106. 5