Download Adding and Subtracting Algebraic Expressions

10.3
Adding and Subtracting
Algebraic Expressions
10.3
OBJECTIVES
1. Identify like terms
2. Combine like terms
To find the perimeter of (or the distance around) a rectangle, we add 2 times the length and
2 times the width. In the language of algebra, this can be written as
L
W
W
Perimeter 2L 2W
L
We call 2L 2W an algebraic expression, or more simply an expression. Recall from
Section 10.1 that an expression allows us to write a mathematical idea in symbols. It can be
thought of as a meaningful collection of letters, numbers, and operation signs.
Some expressions are
1. 5x2
2. 3a 2b
3. 4x3 2y 1
4. 3(x2 y2)
In algebraic expressions, the addition and subtraction signs break the expressions into
smaller parts called terms.
Definitions: Term
A term is a number, or the product of a number and one or more variables,
raised to a power.
In an expression, each sign ( or ) is a part of the term that follows the sign.
Example 1
Identifying Terms
Term
(c) 4x 2y 1 has three terms: 4x3, 2y, and 1.
3
NOTE Notice that each term
“owns” the sign that precedes it.
Term
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(a) 5x2 has one term.
(b) 3a 2b has two terms: 3a and 2b.
Term Term Term
CHECK YOURSELF 1
List the terms of each expression.
(a) 2b4
(b) 5m 3n
(c) 2s2 3t 6
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Note that a term in an expression may have any number of factors. For instance, 5xy is a
term. It has factors of 5, x, and y. The number factor of a term is called the numerical
coefficient. So for the term 5xy, the numerical coefficient is 5.
Example 2
Identifying the Numerical Coefficient
(a)
(b)
(c)
(d)
4a has the numerical coefficient 4.
6a3b4c2 has the numerical coefficient 6.
7m2n3 has the numerical coefficient 7.
Because 1 x x, the numerical coefficient of x is understood to be 1.
CHECK YOURSELF 2
Give the numerical coefficient for each of the following terms.
(b) 5m3n4
(a) 8a2b
(c) y
If terms contain exactly the same letters (or variables) raised to the same powers, they
are called like terms.
Example 3
Identifying Like Terms
(a) The following are like terms.
6a and 7a
5b2 and b2
10x2y3z and 6x2y3z
Each pair of terms has the same letters, with each letter
raised to the same power—the numerical coefficients
can be any number.
3m2 and m2
(b) The following are not like terms.
Different letters
6a and 7b
Different exponents
5b2 and b3
Different exponents
3x2y and 4xy2
CHECK YOURSELF 3
Circle the like terms.
5a2b
ab2
a2b
3a2
4ab
3b2
7a2b
© 2001 McGraw-Hill Companies
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ADDING AND SUBTRACTING ALGEBRAIC EXPRESSIONS
SECTION 10.3
737
Like terms of an expression can always be combined into a single term. Look at the
following:
5x
xxxxxxx
7x
2x
xxxxxxx
Rather than having to write out all those x’s, try
NOTE Here we use the
distributive property from
Section 1.5.
NOTE You don’t have to write
all this out—just do it mentally!
2x 5x (2 5)x 7x
In the same way,
9b 6b (9 6)b 15b
and 10a (4a) (10 4)a 6a
This leads us to the following rule.
Step by Step: Combining Like Terms
To combine like terms, use the following steps.
Step 1 Add or subtract the numerical coefficients.
Step 2 Attach the common variables.
Example 4
Combining Like Terms
Combine like terms.
(a) 8m 5m (8 5)m 13m
(b) 5pq3 4pq3 1pq3 pq3
NOTE Remember that when
(c) 7a3b2 7a3b2 0a3b2 0
any factor is multiplied by 0,
the product is 0.
CHECK YOURSELF 4
Combine like terms.
(a) 6b 8b
(c) 8xy3 7xy3
(b) 12x2 3x2
(d) 9a2b4 9a2b4
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Let’s look at some expressions involving more than two terms. The idea is just the same.
Example 5
Combining Like Terms
NOTE The distributive property
can be used over any number of
like terms.
Combine like terms.
(a) 4xy (xy) 2xy
(4 (1) 2)xy
5xy
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Only like terms can be combined.
(b) 8x (2x) 5y
6x
5y
Like terms
be writing out these steps, but
doing them mentally.
(c) 5m 8n
4m (3n)
(5m 4m) (8n (3n))
9m
Here we have used the associative
and commutative properties.
5n
(d) 4x2 2x (3x2) x
(4x2 (3x2)) (2x x)
x2 3x
As these examples illustrate, combining like terms often means changing the grouping
and the order in which the terms are written. Again all this is possible because of the properties of addition that we introduced in Section 1.2.
CHECK YOURSELF 5
Combine like terms.
(a) 4m2 3m2 8m2
(b) 9ab 3a 5ab
(c) 4p 7q 5p 3q
CHECK YOURSELF ANSWERS
1. (a) 2b4; (b) 5m, 3n; (c) 2s2, 3t, 6
2. (a) 8; (b) 5; (c) 1
3. The like terms are 5a2b, a2b, and 7a2b
4. (a) 14b; (b) 9x2; (c) xy3; (d) 0
5. (a) 9m2; (b) 4ab 3a; (c) 9p 4q
© 2001 McGraw-Hill Companies
NOTE With practice you won’t
Like terms
Name
10.3 Exercises
Section
Date
List the terms of the following expressions.
1. 5a 2
ANSWERS
2. 7a (4b)
1.
3
2
3. 4x
4. 3x
2.
5. 3x2 3x (7)
6. 2a3 (a2) a
3.
4.
Circle the like terms in the following groups of terms.
5.
8. 9m2, 8mn, 5m2, 7m
7. 5ab, 3b, 3a, 4ab
9. 4xy2, 2x2y, 5x2, 3x2y, 5y, 6x2y
10. 8a2b, 4a2, 3ab2, 5a2b, 3ab, 5a2b
6.
7.
8.
9.
Combine the like terms.
11. 3m 7m
10.
12. 6a2 8a2
13. 7b3 10b3
14. 7rs 13rs
15. 21xyz 7xyz
16. 4mn2 15mn2
17. 9z2 (3z 2)
18. 7m (6m)
19. 5a3 (5a3)
20. 13xy (9xy)
21. 19n (18n )
22. 7cd (7cd)
23. 21p2q (6p2q)
24. 17r 3s2 (8r 3s2)
2
2
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
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28.
25. 10x (7x ) 3x
26. 13uv 5uv (12uv)
27. 9a (7a) 4b
28. 5m2 (3m) 6m2
29. 7x 5y (4x) (4y)
30. 6a 11a 7a (9a)
2
2
2
29.
30.
31.
2
2
32.
31. 4a 7b 3 (2a) 3b (2)
32. 5p2 2p 8 4p2 5p (6)
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ANSWERS
33.
33.
2
4
m3 m
3
3
35.
13
3
x2 x 5
5
5
34.
35.
36.
37.
37. 2.3a 7 4.7a 3
34.
1
4
a (2) a
5
5
36.
17
7
y 7 y (3)
12
12
38. 5.8m 4 (2.8m) 11
38.
39.
Perform the indicated operations.
40.
39. The sum of 5a4 and 8a4 is
41.
40. The sum of 9p2 and 12p2 is
42.
43.
41. Subtract 12a3 from 15a3.
44.
45.
42. Subtract 5m3 from 18m3.
46.
43. Subtract 4x from the sum of 8x and 3x.
47.
48.
44. Subtract 8ab from the sum of 7ab and 5ab.
49.
50.
45. Subtract 3mn2 from the sum of 9mn2 and 5mn2.
51.
46. Subtract 4x2y from the sum of 6x2y and 12x2y.
52.
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47. 2(3x 2) 4
48. 3(4z 5) 9
49. 5(6a 2) 12a
50. 7(4w 3) 25w
51. 4s 2(s 4) 4
52. 5p 4(p 3) 8
© 2001 McGraw-Hill Companies
Use the distributive property to remove the parentheses in each expression. Then simplify
by combining like terms.
ANSWERS
Evaluate each of the following expressions if a 2, b 3, and c 5. Be sure to combine
terms when possible as the first step.
53.
54.
55.
56.
53. 7a2 3a
54. 11b2 9b
57.
55. 3c2 5c2
58.
56. 9b3 5b3
59.
57. 5b 3a 2b
58. 7c 2b 3c
59. 5ac2 2ac2
60. 5a3b 2a3b
60.
61.
62.
63.
Using your calculator, evaluate each of the following for the given values of the variables.
Round your answer to the nearest tenth.
64.
61. 7x2 5y3 for x 7.1695 and y 3.128
65.
62. 2x2 3y 5x for x 3.61 and y 7.91
66.
67.
63. 4x2y 2xy2 5x3y for x 1.29 and y 2.56
68.
64. 3x3y 4xy 2x2y2 for x 3.26 and y 1.68
65. Write a paragraph explaining the difference between n2 and 2n.
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66. Complete the explanation: “x3 and 3x are not the same because . . . . ”
67. Complete the statement: “x 2 and 2x are different because . . . . ”
68. Write an English phrase for each algebraic expression below:
(a) 2x3 5x
(b) (2x 5)3
(c) 6(n 4)2
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ANSWERS
69. Work with another student to complete this exercise. Place , , or in the blank in
69.
these statements.
70.
12______21
What happens as the table of numbers is extended? Try more examples.
23______32
What sign seems to occur the most in your table? , , or ?
34______43
45______54
Write an algebraic statement for the pattern of numbers in this table.
Do you think this is a pattern that continues? Add more lines to the
table and extend the pattern to the general case by writing the pattern
in algebraic notation. Write a short paragraph stating your conjecture.
70. Work with other students on this exercise.
n2 1
n2 1
, n,
using odd values of n: 1, 3,
2
2
5, 7, etc. Make a chart like the one below and complete it.
Part 1: Evaluate the three expressions
n
a
n2 1
2
bn
c
n2 1
2
a2
b2
c2
1
3
5
7
9
11
13
15
Answers
1. 5a, 2
3. 4x3
5. 3x2, 3x, 7
7. 5ab, 4ab
9. 2x2y, 3x2y, 6x2y
3
2
11. 10m
13. 17b
15. 28xyz
17. 6z
19. 0
21. n2
2
2
23. 15p q
25. 6x
27. 2a 4b
29. 3x y
31. 2a 10b 1
33. 2m 3
35. 2x 7
37. 7a 10
39. 13a4
41. 3a3
2
43. 7x
45. 11mn
47. 6x 8
49. 42a 10
51. 6s 12
53. 34
55. 200
57. 15
59. 150
61. 206.8
63. 260.6
65.
67.
69.
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© 2001 McGraw-Hill Companies
Part 2: The numbers, a, b, and c that you get in each row have a surprising
relationship to each other. Complete the last three columns and work together to
discover this relationship. You may want to find out more about the history of this
famous number pattern.