Download Unit 4: Factoring (3)

Unit 4: Factoring (3)
4.1 Factoring methods  a. common factors
b. trinomials (trinomial squares) & grouping
c. difference of squares
d. sums and difference of cubes
Factor each polynomial completely. If the polynomial is prime, say so.
1. 3z² + 6z
2. 11x² - 33x³
3. 9ay² - 15a²y
4. 18r²s³ + 12r³s
17. p² - 8p + 9
18. s² - 20s + 36
19. x² + x – 12
20. t² - 2t – 35
21. 3z² + 4z + 1
22. 8 + 2s - s²
23. x² - xy – 30y²
5. 4hk² + 16h²k
6. 16x³ - 64x²
7. 6x²y² + 8x³y
8. y² - 5y
24. u² - 8uv – 12v²
25. 2t² + 5t – 3
26. 3p² - 7p – 6
27. 6x² -7xy – 3y²
28. 2h² + 7hk – 15k²
29. 6x² + 7x -10
30. 4t² - 9t + 6
9. 4ab – 6ac + 12ad
10. 6y² + 3y³
11. 5x²y³ + 15x³y²
12. 24x³ - 36x² + 72x
31. h² - 10h + 24
32. z² - 9z + 12
33. t² + 2t – 15
34. s² - 6s - 27
35. 5v² + 4v – 1
36. 21 – 4x - x²
37. p² + 2pq – 24q²
38. h² - 8hk – 15k²
39. 3x² - 8x + 5
40. 4r² + 8r + 3
41. 6s² + st – 5t²
42. 2u² + uv – 21v²
43. 4y² -17y + 15
44. 2x² + 11x + 12
60. 2x² - 40x + 200
61. 20y² + 100y + 125
62. 12a² + 36a + 27
63. a³ + 24a² + 144a
64. y³ - 18y² + 81y
45. t² + 18t + 81
46. z² - 12z + 36
47. y² - 6y + 9
48. x² - 8x + 16
49. x² + 14x + 49
50. x² + 16x + 64
51. 4y² + 20y + 25
52. 9s² - 24s + 16
53. 121s² - 66st + 9t²
54. 16x² + 40xy + 25y²
55. 25u² - 20u + 4
56. 1 – 8d + 16d²
57. 25y² - 80y + 64
58. 4s² + 4s + 1
59. 32x² + 48x + 18
65. x² - 16
66. y² - 9
67. 9x² - 25
68. 4a² - 49
69. 4x² - 25
70. 100y² - 81
71. 16k² - 1
72. 121b² - 9
73. 16x² - 25
74. 4h² - 91
75. 36p² - 49q²
76. 9x² - 16y²
77. 25a² - 81
78. 9b² - 49
79. 25m² - 36n²
85. uv – u – 2v + 2
86. uv – u – 2 + 2v
87. xy – 2y – x + 2
88. 4ab + 1 – 2a – 2b
89. xy – 3x +2y – 6
90. xy – 3y + 6 – 2y
109. x³ + 8
110. z³ - 1
111. 8a³ + 1
112. p³ - 27
113. 64y³ - 125
91. pq – 2q + 2p – 4
92. ab – 2 – 2b + a
93. ac + ad + bc + bd
94. xy + xz + wy + wz
95. y² - 8y – y + 8
96. 2y² + 6y + 5y + 15
114. 27y³ + 64
115. c³ + 27
116. w³ + 1
117. 27x³ + 1
118. 8 - 27h³
13. 8xy + 10xz – 14xw
14. 10a³ + 15a² - 25a
15. 3xy² + 18x³y
16. 5x - 15x² + 35x³
97. t² + 6t – 2t - 12
98. b³ - b² + 2b – 2
99. x² - y² - 4y – 4
100. u² - v² + 2v – 1
101. z² + 2z + 1 - w²
102. x² - 6x + 9 – 4y²
119. 125p³ + 1
120. a³ - b³
121. y³ - 64
122. x³ + 125
123. d³ - 8
80. 6x² - 6y²
81. 8a² - 8b²
82. st² - s
83. p³q - pq
84. 64u² - 25v²
103. a² + 2ab + b² - 9
104. 2m² + 4mn + 2n² - 50b²
105. x² - 2xy + y² - 25
106. 4y² - 20y + 25 - z²
107. 12x² + 12x + 3 – 3y²
108. a² + 16a + 64 - b²
124. x³ + y³
125. 343a³ + 27
126. 8x³ - 27y³
127. 125p³ - 8q³
128. 64u³ - v³
4.2 Problem Solving using Factoring (Polynomial Equations)
1. The sum of a number and its square is 72. Find the number.
2. The sum of a number and its square is 42. Find the number.
3. The sum of a number and its square is 56. Find the number.
4. Find two consecutive odd integers whose product is 143.
5. Find two consecutive even integers whose product is 168.
6. Find two consecutive integers such that the sum of their squares is 113.
7. Find two consecutive even integers such that the sum of their squares is 340.
8. The sum of the squares of two consecutive odd positive integers is 202. Find the integers.
9. Two positive real numbers have a sum of 5 and product of 5. Find the numbers.
10. A rectangle is 4 cm longer than it is wide, and its area is 117 cm². Find its dimensions.
11. A rectangular garden has perimeter 66 ft and area 216 ft². Find the dimensions of the garden.
12. The area of a right triangle is 44 m². Find the lengths of its legs if one of the legs is 3 m longer than the other.
13. The top of a 15 foot ladder is 3 ft further up the wall than the foot of the ladder is from the bottom of the wall.
How far is the foot of the ladder from the bottom of the wall?
14. The height of a triangle is 7 cm greater than the length of its base and its area is 15 cm². Find the height.
15. The hypotenuse of a right triangle is 25 m long. The length of one leg is 10 m less than twice the other. Find
the length of each leg.
16. The side of a large tent is in the shape of an isosceles triangle whose area is 54 ft² and whose base is 6 ft
shorter than twice its height. Find the height and the base of the side of the tent.
17. Each side of a square is 4 m long. When each side is increased by x m, the area is doubled. Find the value of x.
18. A walkway of uniform width has area 72 m² and surrounds a swimming pool that is 8 m wide and 10 m long.
Find the width of the walkway.
19. Erika has a rectangular picture with dimensions of 3 in. by 5 in. She frames the picture with a border that has a
uniform width of x inches. The framed picture with the border has an area of 63 in². What is x, the width of the
border?
20. A rectangular lot has perimeter 78 ft and area 350 ft². Find the dimensions of the lot.
21. A flower bed is to be 3 m longer than it is wide. The flower bed will have an area of 108 m². What will its
dimensions be?
22. The difference between two positive numbers is 3 and the product is 28. Find the smaller of the two numbers.
4.3 Graphing Quadratics using roots/zeros and y-intercept
Tell whether the ordered pair is a solution of the given equation.
Then give the direction the parabola opens and the parabolas y-intercept.
1. (0, 0); y = - 2x² + 5x - 4
2. (0, 0); y = 4x² + 2x – 7
3. (0, 0); y = - 3x² + 5x – 2
4. (0, 0); y = 2x² + 8x
5. (0, 0); y = - x² - 6
6. (0, 0); y = x² + 3x – 4
For the following quadratic equations: a) state the direction the parabola opens
b) state the y-intercept
c) find the roots/zeros (solutions/x-intercepts) by factoring
d) graph the parabola
7. y = x² - 4
8. y = x² - 2x – 8
9. y = - x² - 2x + 8
10. y = x² - 5x + 4
11. y = x² + 3x
12. y = x² + 2x + 1
13. y = x² + 2
14. y = - x² + 1
15. y = - x² + 8x – 12
16. y = x² + 4x + 3
17. y = x² - 4x + 4
18. y = - 2x² + 10x
19. Which statement is NOT true about the graph
of y = x² - 2x + 9?
20. Which statement is true about the parabola
y = - x² + 4x – 3
A the y-intercept is 9
A the parabola opens up
B (0, 0) is a solution
B x = - 3 and x = - 1 are it’s zeros
C the parabola opens up
C (0, - 2) is a solution
D the graph consists of all points below, but
not on, the parabola.
D the graph is below the line x = 0
Mixed Review Exercises
1. Solve and Graph: │2x - 3│= 9
2. Solve each system: y = - 3x + 5
x – 2y = 4
3. Graph: y ≥ 4
x – 2y ≤ 3
4. Simplify: (- x³ + 3x² - 2x + 2) – (- x³ + 5x² - 8x + 4)
5. Multiply: a. (4x – 5y)²
b. (2x + 3 + 5y)(2x + 3 - 5y)
*Reminder For every journal entry:
1. Date each journal entry with the date it was assigned.
2. Copy the question assigned.
3. Answer the journal questions in complete sentences using your own words and until you feel you have
completely answered the questions. If the journal requires you to solve a problem do so and explain how to
solve the problem if required.
4. MAKE SURE YOU COMPLETE YOUR JOURNALS BEFORE THEY ARE DUE!
Unit 4
1. What is your current grade average? Are you satisfied with this grade? If not, what do you
need to do differently to pull up your grade.
2. a. The constant term c of a trinomial ax² + bx + c is positive. When the trinomial is factored, what is
known about the constants of the factor?
b. The constant term c of a trinomial ax² + bx + c is negative. When the trinomial is factored, what is
known about the constants of the factor?
3. Give the complete factorization to each of the following cases and describe the steps to each
factorization.
a. 3x 2  18 x  27
b. 9 x 2  6 x  1
c. 4a 2  25b 2
d. x 3  27 y 3
e. 8 x 3  125