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 5. 4hk² + 16h²k 9. 4ab – 6ac + 12ad 13. 8xy + 10xz – 14xw
2. 11x² - 33x³ 6. 16x³ - 64x² 10. 6y² + 3y³ 14. 10a³ + 15a² - 25a
3. 9ay² - 15a²y 7. 6x²y² + 8x³y 11. 5x²y³ + 15x³y² 15. 3xy² + 18x³y
4. 18r²s³ + 12r³s 8. y² - 5y 12. 24x³ - 36x² + 72x 16. 5x - 15x² + 35x³
17. p² - 8p + 924. u² - 8uv – 12v²31. h² - 10h + 24 38. h² - 8hk – 15k²
18. s² - 20s + 36 25. 2t² + 5t – 3 32. z² - 9z + 12 39. 3x² - 8x + 5
19. x² + x – 12 26. 3p² - 7p – 6 33. t² + 2t – 15 40. 4r² + 8r + 3
20. t² - 2t – 35 27. 6x² -7xy – 3y² 34. s² - 6s - 27 41. 6s² + st – 5t²
21. 3z² + 4z + 1 28. 2h² + 7hk – 15k² 35. 5v² + 4v – 1 42. 2u² + uv – 21v²
22. 8 + 2s - s² 29. 6x² + 7x -10 36. 21 – 4x - x² 43. 4y² -17y + 15
23. x² - xy – 30y² 30. 4t² - 9t + 6 37. p² + 2pq – 24q² 44. 2x² + 11x + 12
45. t² + 18t + 81 50. x² + 16x + 64 55. 25u² - 20u + 4 60. 2x² - 40x + 200
46. z² - 12z + 36 51. 4y² + 20y + 25 56. 1 – 8d + 16d² 61. 20y² + 100y + 125
47. y² - 6y + 9 52. 9s² - 24s + 16 57. 25y² - 80y + 64 62. 12a² + 36a + 27
48. x² - 8x + 16 53. 121s² - 66st + 9t² 58. 4s² + 4s + 1 63. a³ + 24a² + 144a
49. x² + 14x + 49 54. 16x² + 40xy + 25y² 59. 32x² + 48x + 18 64. y³ - 18y² + 81y
65. x² - 16 70. 100y² - 81 75. 36p² - 49q²80. 6x² - 6y²
66. y² - 9 71. 16k² - 1 76. 9x² - 16y²81. 8a² - 8b²
67. 9x² - 25 72. 121b² - 9 77. 25a² - 8182. st² - s
68. 4a² - 49 73. 16x² - 25 78. 9b² - 4983. p³q - pq
69. 4x² - 25 74. 4h² - 91 79. 25m² - 36n²84. 64u² - 25v²
85. uv – u – 2v + 291. pq – 2q + 2p – 497. t² + 6t – 2t - 12 103. a² + 2ab + b² - 9
86. uv – u – 2 + 2v92. ab – 2 – 2b + a98. b³ - b² + 2b – 2 104. 2m² + 4mn + 2n² - 50b²
87. xy – 2y – x + 293. ac + ad + bc + bd99. x² - y² - 4y – 4 105. x² - 2xy + y² - 25
88. 4ab + 1 – 2a – 2b94. xy + xz + wy + wz100. u² - v² + 2v – 1 106. 4y² - 20y + 25 - z²
89. xy – 3x +2y – 695. y² - 8y – y + 8101. z² + 2z + 1 - w² 107. 12x² + 12x + 3 – 3y²
90. xy – 3y + 6 – 2y96. 2y² + 6y + 5y + 15102. x² - 6x + 9 – 4y² 108. a² + 16a + 64 - b²
109. x³ + 8 114. 27y³ + 64 119. 125p³ + 1124. x³ + y³
110. z³ - 1 115. c³ + 27 120. a³ - b³125. 343a³ + 27
111. 8a³ + 1 116. w³ + 1 121. y³ - 64126. 8x³ - 27y³
112. p³ - 27 117. 27x³ + 1 122. x³ + 125127. 125p³ - 8q³
113. 64y³ - 125 118. 8 - 27h³ 123. d³ - 8128. 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.
4.3Graphing 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 – 73. (0, 0); y = - 3x² + 5x – 2
4. (0, 0); y = 2x² + 8x5. (0, 0); y = - x² - 66. (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 - 48. y = x² - 2x – 89. y = - x² - 2x + 8
10. y = x² - 5x + 411. y = x² + 3x12. y = x² + 2x + 1
13. y = x² + 214. y = - x² + 115. y = - x² + 8x – 12
16. y = x² + 4x + 317. y = x² - 4x + 418. y = - 2x² + 10x
19. Which statement is NOT true about the graph 20. Which statement is true about the parabola
of y = x² - 2x + 9? 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 D the graph is below the line x = 0
not on, the parabola.
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.
- 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?
- Give the complete factorization to each of the following cases and describe the steps to each factorization.