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EXAMPLE
10
Use the distributive law to factor each of the following.
a) 3x + 3y
b) 7x + 21y + 7
SOLUTION
a) By the distributive law,
3x + 3y = 31x + y2.
The common factor for 3x and 3y is 3.
b) 7x + 21y + 7 = 7 # x + 7 # 3y + 7 # 1
= 71x + 3y + 12
The common factor is 7.
Using the distributive law.
Be sure to include both the 1 and
the common factor, 7.
TRY EXERCISE
69
To check our factoring, we multiply to see if the original expression is
obtained. For example, to check the factorization in Example 10(b), note that
71x + 3y + 12 = 7 # x + 7 # 3y + 7 # 1
= 7x + 21y + 7.
Since 7x + 21y + 7 is what we started with in Example 10(b), we have a check.
CAUTION! Do not confuse terms with factors. Terms are separated by plus
signs, and factors are parts of products. The distributive law is used when
there are two or more terms inside parentheses. For example, in the expression a1b # c2, b and c are factors, not terms. We can use the commutative and
associative laws to reorder and regroup the factors, but the distributive law
does not apply here. Thus,
a1b # c2 = a # b # a # c
1.2
EXERCISE SET
i
Concept Reinforcement Complete each sentence
using one of these terms: commutative, associative, or
distributive.
1. 8 + t is equivalent to t + 8 by the
law for addition.
2. 31xy2 is equivalent to 13x2y by the
law for multiplication.
3. 15b2c is equivalent to 51bc2 by the
law for multiplication.
4. mn is equivalent to nm by the
law for multiplication.
5. x1y + z2 is equivalent to xy + xz by the
law.
but
a1b # c2 = 1a # b2 # c.
For Extra Help
6. 19 + a2 + b is equivalent to 9 + 1a + b2 by the
law for addition.
7. a + 16 + d2 is equivalent to 1a + 62 + d by the
law for addition.
8. 31t + 42 is equivalent to 314 + t2 by the
law for addition.
9. 51x + 22 is equivalent to 1x + 225 by the
law for multiplication.
10. 21a + b2 is equivalent to 2 # a + 2 # b by the
law.
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1. 2
Use the commutative law of addition to write an equivalent expression.
11. 11 + t
12. a + 2
13. 4 + 8x
14. ab + c
15. 9x + 3y
16. 3a + 7b
17. 51a + 12
18. 91x + 52
Use the commutative law of multiplication to write an
equivalent expression.
19. 7x
20. xy
21. st
22. 13m
23. 5 + ab
24. x + 3y
25. 51a + 12
26. 91x + 52
Use the associative law of addition to write an equivalent
expression.
27. 1x + 82 + y
28. 15 + m2 + r
T H E C O M M U TAT I V E , A S S O C I AT I V E , A N D D I S T R I B U T I V E L AW S
Multiply.
47. 21x + 152
48. 31x + 52
49. 411 + a2
50. 61v + 42
51. 813 + y2
52. 71s + 12
53. 1019x + 62
54. 916m + 72
55. 51r + 2 + 3t2
56. 415x + 8 + 3p2
57. 1a + b22
58. 1x + 227
59. 1x + y + 225
List the terms in each expression.
61. x + xyz + 1
62. 9 + 17a + abc
a
+ 5b
63. 2a +
3b
64. 3xy + 20 +
29. u + 1v + 72
65. 41x + y2
31. 1ab + c2 + d
67. 4x + 4y
30. x + 12 + y2
60. 12 + a + b26
4a
b
66. 17 + y22
32. 1m + np2 + r
68. 14 + 2y
Use the associative law of multiplication to write an
equivalent expression.
Use the distributive law to factor each of the following.
Check by multiplying.
33. 18x2y
34. 14u2v
35. 21ab2
36. 917r2
69. 2a + 2b
70. 5y + 5z
37. 3321a + b24
38. 53x12 + y24
71. 7 + 7y
72. 13 + 13x
73. 32x + 4
74. 20a + 5
39. s + 1t + 62
75. 5x + 10 + 15y
76. 3 + 27b + 6c
77. 7a + 35b
78. 8x + 24y
41. 117a2b
79. 44x + 11y + 22z
80. 14a + 56b + 7
Use the commutative and/or associative laws to write two
equivalent expressions. Answers may vary.
40. 7 + 1v + w2
42. x13y2
Use the commutative and/or associative laws to show why
the expression on the left is equivalent to the expression on
the right.Write a series of steps with labels, as in Example 4.
List the factors in each expression.
81. 5n
82. uv
83. 31x + y2
43. 11 + x2 + 2 is equivalent to x + 3
84. 1a + b212
85. 7 # a # b
86. m # n # 2
45. 1m # 327 is equivalent to 21m
87. 1a - b21x - y2
88. 13 - a21b + c2
89. Is subtraction commutative? Why or why not?
44. 12a24 is equivalent to 8a
46. 4 + 19 + x2 is equivalent to x + 13
90. Is division associative? Why or why not?
19
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Skill Review
To the student and the instructor: Exercises included for
Skill Review include skills previously studied in the text.
Often these exercises provide preparation for the next
section of the text. The numbers in brackets immediately
following the directions or exercise indicate the section
in which the skill was introduced. The answers to all
Skill Review exercises appear at the back of the book.
If a Skill Review exercise gives you difficulty, review the
material in the indicated section of the text.
Translate to an algebraic expression. [1.1]
91. Half of Kara’s salary
92. Twice the sum of m and 3
Synthesis
93. Give an example illustrating the distributive law,
and identify the terms and the factors in your example. Explain how you can determine terms and
factors in an expression.
94. Explain how the distributive, commutative, and
associative laws can be used to show that
213x + 4y2 is equivalent to 6x + 8y.
Tell whether the expressions in each pairing are equivalent. Then explain why or why not.
95. 8 + 41a + b2 and 412 + a + b2
96. 51a # b2 and 5 # a # 5 # b
97. 7 , 3m and m # 3 , 7
98. 1rt + st25 and 5t1r + s2
99. 30y + x # 15 and 5321x + 3y24
100. 3c12 + 3b245 and 10c + 15bc
101. Evaluate the expressions 312 + x2 and 6 + x for
x = 0. Do your results indicate that 312 + x2 and
6 + x are equivalent? Why or why not?
102. Factor 15x + 40. Then evaluate both 15x + 40
and the factorization for x = 4. Do your results
guarantee that the factorization is correct? Why or
why not? (Hint: See Exercise 101.)
Mental Addition
Focus: Application of commutative and
associative laws
Time: 10 minutes
Group size: 2–3
Legend has it that while still in grade school,
the mathematician Carl Friedrich Gauss
(1777–1855) was able to add the numbers from
1 to 100 mentally. Gauss did not add them
sequentially, but rather paired 1 with 99, 2
with 98, and so on.
ACTIVITY
1. Use a method similar to Gauss’s to simplify the
following:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
One group member should add from left to
right as a check.
2. Use Gauss’s method to find the sum of the
first 25 counting numbers:
1 + 2 + 3 + Á + 23 + 24 + 25.
Again, one student should add from left to
right as a check.
3. How were the associative and commutative laws
applied in parts (1) and (2) above?
4. Now use a similar approach involving both
addition and division to find the sum of the
first 10 counting numbers:
1 + 2 + 3 + Á + 10
+ 10 + 9 + 8 + Á + 1
5. Use the approach in step (4) to find the sum of
the first 100 counting numbers. Are the associative and commutative laws applied in this
method, too? How is the distributive law used
in this approach?