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3. Lesson 9-1 Factors and Greatest Common Factors
Lesson 9-2 Factoring Using the Distributive Property
Lesson 9-3 Factoring Trinomials: x2 + bx + c
Lesson 9-4 Factoring Trinomials: ax2 + bx + c
Lesson 9-5 Factoring Differences of Squares
Lesson 9-6 Perfect Squares and Factoring
Contents

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

5. 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.
Example 1-1a

6. 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.
Example 1-1a

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

8. 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
Example 1-2a

9. 84 21 4 3 7 2 2 Method 2 Use a factor tree. and
21 4
3 7
2 2
and
All of the factors in the last branch of the factor tree are prime.
Answer: Thus, the prime factorization of 84 is or
Example 1-2a

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

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

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

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

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

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

16. Circle the common prime factors.
Find the GCF of 12 and 18.
Factor each number.
Circle the common prime factors.
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.
Answer: The GCF of 12 and 18 is 6.
Example 1-5a

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

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

19. Crafts Rene has crocheted 32 squares for an afghan
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.
Example 1-6a

20. Answer: The maximum amount of ribbon Rene will need is 66 feet.
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.
Example 1-6a

21. 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
Example 1-6b

22. End of Lesson 1

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

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

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

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

27. Use the Distributive Property to factor each polynomial. a.
b. 6ab a2b2 = 27ab3
Answer:
Answer: 3ab2 (2 + 5a + 9b)
Example 2-1b

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

29. Factor
Answer:
Example 2-2b

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

31. Factor
Answer:
Example 2-3b

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

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

34. Solve Then check the solutions.
Answer: {3, –2}
Example 2-4b

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

36. Answer: The solution set is Check by substituting
0 and for y in the original equation.
Example 2-5a

37. Solve
Answer:
Example 2-5b

38. End of Lesson 2

39. 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 5 Solve an Equation by Factoring
Example 6 Solve a Real-World Problem by Factoring
Lesson 3 Contents

40. 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
Example 3-1a

41. Check You can check the result by multiplying the two factors.
FOIL method F O I L
Simplify.
Example 3-1a

42. Factor
Answer:
Example 3-1b

43. Factor
In this trinomial, and This means is 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
Example 3-2a

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

45. Factor
Answer:
Example 3-2b

46. The correct factors are –3 and 6.
In this trinomial, and This means is 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, , –9 –2, , –6 –3, 6
– – –
The correct factors are –3 and 6.
Example 3-3a

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

48. Factor
Answer:
Example 3-3b

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

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

51. Factor
Answer:
Example 3-4b

52. Rewrite the equation so that one side equals 0.
Solve
Check 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
Example 3-5a

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

54. Solve
Check your solutions.
Answer:
Example 3-5b

55. 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?
Explore Begin by making a diagram like the one shown to the right, labeling the appropriate dimensions.
Example 3-6a

56. Plan Let the 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.
Example 3-6a

57. Factor. or Zero Product Property Solve each equation.
Examine The solution set is Only 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.
Example 3-6a

58. Photography Adina has a. photograph
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?
Answer:
Example 3-6b

59. End of Lesson 3

60. Example 1 Factor ax2 + bx + c
Example 2 Factor When a, b, and c Have a Common Factor
Example 3 Determine Whether a Polynomial Is Prime
Example 4 Solve Equations by Factoring
Example 5 Solve Real-World Problems by Factoring
Lesson 4 Contents

61. 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.
Example 4-1a

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

63. Factor
Answer:
Example 4-1a

64. 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 –3, –24 –4, –18
Sum of Factors
Factors of 72
The correct factors are –4, –18.
Example 4-1b

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

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

67. 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 factor Since the lead coefficient is 1, find the two factors of 8 whose sum is 6.
9 6
1, 8 2, 4
Sum of Factors
Factors of 8
The correct factors are 2 and 4.
Example 4-2a

68. 8
Answer: So, Thus, the complete factorization of is
Example 4-2a

69. Factor
Answer:
Example 4-2b

70. 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.
14 –14 2 –2
–1, 15 1, –15 –3, 5 3, –5
Sum of Factors
Factors of –15
Example 4-3a

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

72. Factor
Answer: prime
Example 4-3b

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

74. Solve
Answer:
Example 4-4b

75. 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
Example 4-5a

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

77. The solutions are and seconds. 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
Example 4-5a

78. 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
Example 4-5b

79. End of Lesson 4

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

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

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

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

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

85. Factor
Answer:
Example 5-2b

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

87. Factor
Answer:
Example 5-3b

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

89. Factor
Answer:
Example 5-4b

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

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

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

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

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

95. b. What value of x will result in a figure
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.
Example 5-6a

96. 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.
Answer:
b. Find x so that A is the area of the original triangle,
Translate the verbal statement.
Example 5-6a

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

98. b. What value of x will result in a figure
Extended-Response Test Item A square with side length x is cut from the larger square 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 of the area of the
original square?
Answer:
Answer: 3
Example 5-6b

99. End of Lesson 5

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

101. 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 ?
Yes,
Answer: is a perfect square trinomial.
Write as
Factor using the pattern.
Example 6-1a

102. 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.
Example 6-1a

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

104. 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:
Example 6-2a

105. 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 form Are 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.
Example 6-2a

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

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

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

109. Solve
Answer:
Example 6-3b

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

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

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

113. Subtract 9 from each side.
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 are or about –6.17 and
or about –11.83.
Example 6-4a

114. Check You can check your answer using a graphing calculator. Graph and Using the INTERSECT feature of your graphing calculator, find where The check of –6.17 as one of the approximate solutions is shown.
Example 6-4a

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

116. End of Lesson 6

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