2. Welcome to Interactive Chalkboard Algebra 1 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 9-1Factors and Greatest Common Factors Lesson 9-2Factoring Using the Distributive Property Lesson 9-3Factoring Trinomials: x 2 + bx + c Lesson 9-4Factoring Trinomials: ax 2 + bx + c Lesson 9-5Factoring Differences of Squares Lesson 9-6Perfect Squares and Factoring

5. Lesson 1 Contents Example 1Classify Numbers as Prime or Composite Example 2Prime Factorization of a Positive Integer Example 3Prime Factorization of a Negative Integer Example 4Prime Factorization of a Monomial Example 5GCF of a Set of Monomials Example 6Use Factors

6. Example 1-1a Factor 22. Then classify it as prime or composite. To find the factors of 22, list all pairs of whole numbers whose product is 22. Answer: Since 22 has more than two factors, it is a composite number. The factors of 22, in increasing order, are 1, 2, 11, and 22.

7. Example 1-1a Factor 31. Then classify it as prime or composite. The only whole numbers that can be multiplied together to get 31 are 1 and 31. Answer:The factors of 31 are 1 and 31. Since the only factors of 31 are 1 and itself, 31 is a prime number.

8. Example 1-1b Factor each number. Then classify it as prime or composite. a. 17 b. 25 Answer: 1, 17 ; prime Answer: 1, 5, 25 ; composite

9. Example 1-2a Find the prime factorization of 84. Method 1 The least prime factor of 84 is 2. The least prime factor of 42 is 2. The least prime factor of 21 is 3. All of the factors in the last row are prime. Answer: Thus, the prime factorization of 84 is

10. Example 1-2a Method 2 Use a factor tree. 84 214 37372 and All of the factors in the last branch of the factor tree are prime. Answer: Thus, the prime factorization of 84 is or

11. Example 1-2b Find the prime factorization of 60. Answer: or

12. Example 1-3a Find the prime factorization of –132. Express –132 as –1 times 132. Answer: The prime factorization of –132 is or / \

13. Example 1-3b Find the prime factorization of –154. Answer:

14. Example 1-4a Factor completely. Answer:in factored form is

15. Example 1-4a Factor completely. Answer:in factored form is Express –26 as –1 times 26.

16. Example 1-4b Factor each monomial completely. a. b. Answer:

17. Example 1-5a Find the GCF of 12 and 18. The integers 12 and 18 have one 2 and one 3 as common prime factors. The product of these common prime factors, or 6, is the GCF. Factor each number. Circle the common prime factors. Answer:The GCF of 12 and 18 is 6.

18. Factor each number. Example 1-5a Circle the common prime factors. Find the GCF of. Answer:The GCF of andis.

19. Example 1-5b Find the GCF of each set of monomials. a. 15 and 35 b. and Answer: 5 Answer:

20. Example 1-6a Crafts Rene has crocheted 32 squares for an afghan. Each square is 1 foot square. She is not sure how she will arrange the squares but does know it will be rectangular and have a ribbon trim. What is the maximum amount of ribbon she might need to finish an afghan? Find the factors of 32 and draw rectangles with each length and width. Then find each perimeter. The factors of 32 are 1, 2, 4, 8, 16, 32.

21. Example 1-6a The greatest perimeter is 66 feet. The afghan with this perimeter has a length of 32 feet and a width of 1 foot. Answer: The maximum amount of ribbon Rene will need is 66 feet.

22. Example 1-6b Mary wants to plant a rectangular flower bed in her front yard with a stone border. The area of the flower bed will be 45 square feet and the stones are one foot square each. What is the maximum number of stones that Mary will need to go around all four sides of the flower bed? Answer: 92 feet

23. End of Lesson 1

24. Lesson 2 Contents Example 1Use the Distributive Property Example 2Use Grouping Example 3Use the Additive Inverse Property Example 4Solve an Equation in Factored Form Example 5Solve an Equation by Factoring

25. Example 2-1a Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. Use the Distributive Property to factor. First, find the CGF of 15x and. Factor each number. Circle the common prime factors. GFC: Rewrite each term using the GCF. Simplify remaining factors. Distributive Property

26. Example 2-1a Answer: The completely factored form of is

27. Example 2-1a Factor each number. Circle the common prime factors. Use the Distributive Property to factor. Rewrite each term using the GCF. Distributive Property Answer: The factored form of is GFC: or

28. Example 2-1b Use the Distributive Property to factor each polynomial. a. b. Answer:

29. Example 2-2a Factor Group terms with common factors. Factor the GCF from each grouping. Answer: Distributive Property

30. Example 2-2b Factor Answer:

31. Example 2-3a Factor Group terms with common factors. Factor GCF from each grouping. Answer: Distributive Property

32. Example 2-3b Factor Answer:

33. Example 2-4a SolveThen check the solutions. If, then according to the Zero Product Property eitheror Original equation or Set each factor equal to zero. Solve each equation. Answer: The solution set is

34. Example 2-4a Check Substitute 2 and for x in the original equation.

35. Example 2-4b SolveThen check the solutions. Answer: {3, –2}

36. Example 2-5a SolveThen check the solutions. Write the equation so that it is of the form Original equation Solve each equation. Factor the GCF of 4y and which is 4y. Zero Product Property or Subtract from each side.

37. Example 2-5a Answer: The solution set isCheck by substituting 0 andfor y in the original equation.

38. Example 2-5b Solve Answer:

39. End of Lesson 2

40. Lesson 3 Contents Example 1 b and c Are Positive Example 2 b Is Negative and c Is Positive Example 3 b Is Positive and c Is Negative Example 4 b Is Negative and c Is Negative Example 5Solve an Equation by Factoring Example 6Solve a Real-World Problem by Factoring

41. Example 3-1a Factor In this trinomial,and You need to find the two numbers whose sum is 7 and whose product is 12. Make an organized list of the factors of 12, and look for the pair of factors whose sum is 7. Factors of 12 Sum of Factors 1, 12 2, 6 3, 4 13 8 7 The correct factors are 3 and 4. Write the pattern. Answer: and

42. Example 3-1a Check You can check the result by multiplying the two factors. FOIL method FOIL Simplify.

43. Example 3-1b Factor Answer:

44. Example 3-2a Factor In this trinomial,and This meansis negative and mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 27, and look for the pair whose sum is –12. Factors of 27 Sum of Factors –1, –27 –3, –9 –28 –12 The correct factors are –3 and –9. Write the pattern. Answer: and

45. Example 3-2a Check You can check this result by using a graphing calculator. Graphand on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly.

46. Example 3-2b Factor Answer:

47. Example 3-3a Factor In this trinomial,and This meansis positive and mn is negative, so either m or n is negative, but not both. Therefore, make a list of the factors of –18 where one factor of each pair is negative. Look for the pair of factors whose sum is 3. Factors of –18 Sum of Factors 1, –18 –1, 18 2, –9 –2, 9 3, –6 –3, 6 –17 17 – 7 7 – 3 3 The correct factors are –3 and 6.

48. Example 3-3a Write the pattern. Answer: and

49. Example 3-3b Factor Answer:

50. Example 3-4a Factors of –20 Sum of Factors 1, –20 –1, 20 2, –10 –2, 10 4, –5 –4, 5 –19 19 – 8 8 – 1 1 The correct factors are 4 and –5. Factor Sinceand is negative and mn is negative. So either m or n is negative, but not both.

51. Example 3-4a Write the pattern. Answer: and

52. Example 3-4b Factor Answer:

53. Example 3-5a SolveCheck your solutions. Original equation Rewrite the equation so that one side equals 0. Factor. or Zero Product Property Solve each equation. Answer: The solution is

54. Example 3-5a Check Substitute –5 and 3 for x in the original equation.

55. Example 3-5b SolveCheck your solutions. Answer:

56. Example 3-6a Architecture Marion has a small art studio measuring 10 feet by 12 feet in her backyard. She wants to build a new studio that has three times the area of the old studio by increasing the length and width by the same amount. What will be the dimensions of the new studio? ExploreBegin by making a diagram like the one shown to the right, labeling the appropriate dimensions.

57. Example 3-6a PlanLetthe amount added to each dimension of the studio. The new length times the new width equals the new area. old area Solve Write the equation. Multiply. Subtract 360 from each side.

58. Example 3-6a Factor. Solve each equation. Zero Product Property or ExamineThe solution set isOnly 8 is a valid solution, since dimensions cannot be negative. Answer: The length of the new studio should be or 20 feet and the new width should be or 18 feet.

59. Photography Adina has a photograph. She wants to enlarge the photograph by increasing the length and width by the same amount. What dimensions of the enlarged photograph will be twice the area of the original photograph? Example 3-6b Answer:

60. End of Lesson 3

61. Lesson 4 Contents Example 1Factor ax 2 + bx + c Example 2Factor When a, b, and c Have a Common Factor Example 3Determine Whether a Polynomial Is Prime Example 4Solve Equations by Factoring Example 5Solve Real-World Problems by Factoring

62. Example 4-1a Factor In this trinomial,and You need to find two numbers whose sum is 27 and whose product is or 50. Make an organized list of factors of 50 and look for the pair of factors whose sum is 27. Factors of 50 Sum of Factors 1, 50 2, 25 51 27 The correct factors are 2 and 25.

63. Example 4-1a Write the pattern. Group terms with common factors. Factor the GCF from each grouping. and Answer: Distributive Property Check You can check this result by multiplying the two factors. FOIL method FOIL Simplify.

64. Example 4-1a Factor Answer:

65. The correct factors are –4, –18. Example 4-1b Factor In this trinomial,and Since b is negative, is negative. Since c is positive, mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of or 72, and look for the pair of factors whose sum is –22. –73 –38 –27 –22 –1, –72 –2, –36 –4, –24 –4, –18 Sum of Factors Factors of 72

66. Example 4-1b Write the pattern. and Group terms with common factors. Factor the GCF from each grouping. Distributive Property Answer:

67. Example 4-1b a. Factor b. Factor Answer:

68. Example 4-2a 9696 1, 8 2, 4 Sum of Factors Factors of 8 Factor Notice that the GCF of the terms, and 32 is 4. When the GCF of the terms of a trinomial is an integer other than 1, you should first factor out this GCF. Distributive Property Now factorSince the lead coefficient is 1, find the two factors of 8 whose sum is 6. The correct factors are 2 and 4.

69. Example 4-2a Answer: So,Thus, the complete factorization ofis

70. Example 4-2b Factor Answer:

71. Example 4-3a 14 –14 2 –2 –1, 15 1, –15 –3, 5 3, –5 Sum of Factors Factors of –15 Factor In this trinomial,and Since b is positive, is positive. Since c is negative, mn is negative, so either m or n is negative, but not both. Therefore, make a list of all the factors of 3(–5) or –15, where one factor in each pair is negative. Look for the pair of factors whose sum is 7.

72. Example 4-3a There are no factors whose sum is 7. Therefore, cannot be factored using integers. Answer:is a prime polynomial.

73. Example 4-3b Factor Answer: prime

74. Example 4-4a Solve Original equation Rewrite so one side equals 0. Factor the left side. orZero Product Property Solve each equation. Answer: The solution set is

75. Example 4-4b Solve Answer:

76. Example 4-5a Model Rockets Ms. Nguyen’s science class built an air-launched model rocket for a competition. When they test-launched their rocket outside the classroom, the rocket landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight? Use the equation

77. Example 4-5a Vertical motion model Subtract 30 from each side. Factor out –4. Divide each side by –4. Factor orZero Product Property Solve each equation.

78. Example 4-5a The solutions areandseconds. The first time represents how long it takes the rocket to reach a height of 30 feet on its way up. The second time represents how long it will take for the rocket to reach the height of 30 feet again on its way down. Thus the rocket will be in flight for 3.5 seconds before coming down again. Answer: 3.5 seconds

79. Example 4-5b When Mario jumps over a hurdle, his feet leave the ground traveling at an initial upward velocity of 12 feet per second. Find the time t in seconds it takes for Mario’s feet to reach the ground again. Use the equation Answer: second

80. End of Lesson 4

81. Lesson 5 Contents Example 1Factor the Difference of Squares Example 2Factor Out a Common Factor Example 3Apply a Factoring Technique More Than Once Example 4Apply Several Different Factoring Techniques Example 5Solve Equations by Factoring Example 6Use Differences of Two Squares

82. Example 5-1a Factor. Write in form Answer: Factor the difference of squares.

83. Example 5-1a Factor. Answer: Factor the difference of squares. and

84. Factor each binomial. a. b. Example 5-1b Answer:

85. Example 5-2a Factor The GCF ofand 27b is 3b. and Answer: Factor the difference of squares.

86. Example 5-2b Answer: Factor

87. Example 5-3a Factor The GCF of and 2500 is 4. and Factor the difference of squares. and Factor the difference of squares. Answer:

88. Example 5-3b Factor Answer:

89. Example 5-4a is the common factor. Factor the difference of squares, into. Answer: Factor Original Polynomial Factor out the GCF. Group terms with common factors. Factor each grouping.

90. Example 5-4b Factor Answer:

91. Example 5-5a Solveby factoring. Check your solutions. Original equation. and Factor the difference of squares. or Zero Product Property Solve each equation.

92. Example 5-5a Answer: The solution set is Check each solution in the original equation.

93. Example 5-5a Solveby factoring. Check your solutions. Original equation Subtract 3y from each side. The GCF of and 3y is 3y. and

94. Example 5-5a Answer: The solution set is Check each solution in the original equation. Applying the Zero Product Property, set each factor equal to zero and solve the resulting three equations. or

95. Answer: Solve each equation by factoring. Check your solutions. a. b. Example 5-5b Answer:

96. Example 5-6a Extended-Response Test Item A square with side length x is cut from a right triangle shown below. a. Write an equation in terms of x that represents the area A of the figure after the corner is removed. b.What value of x will result in a figure that is the area of the original triangle? Show how you arrived at your answer.

97. b. Find x so that A is the area of the original triangle, Example 5-6a Read the Test Item A is the area of the triangle minus the area of the square that is to be removed. Solve the Test Item a. The area of the triangle is or 64 square units and the area of the square is square units. Translate the verbal statement. Answer:

98. Example 5-6a Subtract 48 from each side. and Simplify. Factor the difference of squares. Simplify. or Zero Product Property Answer: Since length cannot be negative, the only reasonable solution is 4. Solve each equation.

99. a.Write an equation in terms of x that represents the area A of the figure after the corner is removed. b.What value of x will result in a figure that is of the area of the original square? Example 5-6b Extended-Response Test Item A square with side length x is cut from the larger square shown below. Answer: Answer: 3

100. End of Lesson 5

101. Lesson 6 Contents Example 1Factor Perfect Square Trinomials Example 2Factor Completely Example 3Solve Equations with Repeated Factors Example 4Use the Square Root Property to Solve Equations

102. Example 6-1a Determine whether is a perfect square trinomial. If so, factor it. Answer:is a perfect square trinomial. 3. Is the middle term equal to? Yes, 1. Is the first term a perfect square? Yes, 2. Is the last term a perfect square?Yes, Write as Factor using the pattern.

103. Example 6-1a Determine whether is a perfect square trinomial. If so, factor it. 1. Is the first term a perfect square? Yes, 2. Is the last term a perfect square?Yes, 3. Is the middle term equal to? No, Answer:is not a perfect square trinomial.

104. Example 6-1b Determine whether each trinomial is a perfect square trinomial. If so, factor it. a. b. Answer: not a perfect square trinomial Answer: yes;

105. Example 6-2a Factor. First check for a GCF. Then, since the polynomial has two terms, check for the difference of squares. 6 is the GCF. and Factor the difference of squares. Answer:

106. Example 6-2a Factor. This polynomial has three terms that have a GCF of 1. While the first term is a perfect square, the last term is not. Therefore, this is not a perfect square trinomial. This trinomial is in the formAre there two numbers m and n whose product is and whose sum is 8 ? Yes, the product of 20 and –12 is –240 and their sum is 8.

107. Example 6-2a Write the pattern. and Group terms with common factors. Factor out the GCF from each grouping. is the common factor. Answer:

108. Example 6-2b Factor each polynomial. a. b. Answer:

109. Example 6-3a Solve Recognize as a perfect square trinomial. Original equation Factor the perfect square trinomial. Set the repeated factor equal to zero. Solve for x. Answer: Thus, the solution set isCheck this solution in the original equation.

110. Example 6-3b Solve Answer:

111. Example 6-4a Solve. Original equation Square Root Property Add 7 to each side. Simplify. Separate into two equations. or Answer: The solution set isCheck each solution in the original equation.

112. Example 6-4a Solve. Original equation Recognize perfect square trinomial. Factor perfect square trinomial. Square Root Property Subtract 6 from each side.

113. Example 6-4a Answer: The solution set isCheck this solution in the original equation. or Separate into two equations. Simplify.

114. Example 6-4a Solve. Original equation Square Root Property Subtract 9 from each side. Answer: Since 8 is not a perfect square, the solution set is Using a calculator, the approximate solutions areor about –6.17 and or about –11.83.

115. Example 6-4a Check You can check your answer using a graphing calculator. GraphandUsing the INTERSECT feature of your graphing calculator, find whereThe check of –6.17 as one of the approximate solutions is shown.

116. Solve each equation. Check your solutions. a. b c. Example 6-4b Answer:

117. End of Lesson 6

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