Download Adding and Subtracting Terms

1.6
Adding and Subtracting Terms
1.6
OBJECTIVES
1.
2.
3.
4.
Identify terms and like terms
Combine like terms
Add algebraic expressions
Subtract algebraic expressions
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 1.1 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
3a 2b
5x 2
4x3 (2y) 1
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
(a) 5x2 has one term.
(b) 3a 2b has two terms: 3a and 2b.
Term Term
© 2001 McGraw-Hill Companies
Term Term
(c) 4x 3 (2y) 1 has three terms: 4x3, 2y, and 1.
written as 4x3 2y 1
NOTE This could also be
Term
CHECK YOURSELF 1
List the terms of each expression.
(a) 2b4
(b) 5m 3n
(c) 2s2 3t 6
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.
115
THE LANGUAGE OF ALGEBRA
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
3m2 and m2
Each pair of terms has the same letters, with each letter
raised to the same power—the numerical coefficients can be
any number.
(b) The following are not like terms.
Different letters
6a and 7b
Different exponents
5b2 and b3
Different exponents
CHECK YOURSELF 3
Circle the like terms.
5a2b
ab2
a2b
3a2
4ab
3b2
7a 2b
Like terms of an expression can always be combined into a single term. Look at the
following:
5x
xxxxxxx
7x
2x
xxxxxxx
© 2001 McGraw-Hill Companies
3x 2y and 4xy 2
CHAPTER 1
116
ADDING AND SUBTRACTING TERMS
SECTION 1.6
117
Rather than having to write out all those x’s, try
NOTE Here we use the
2x 5x (2 5)x 7x
distributive property from
Section 1.2.
In the same way,
NOTE You don’t have to write
9b 6b (9 6)b 15b
all this out—just do it mentally!
and 10a (4a) (10 (4))a 6a
This leads us to the following rule.
Step by Step:
To Combine Like Terms
To combine like terms, use the following steps.
Step 1
Step 2
Add or subtract the numerical coefficients.
Attach the common variables.
Example 4
Combining Like Terms
Combine like terms.*
(a) 8m 5m (8 5)m 13m
(b) 5pq3 4pq3 5pq3 (4pq3) 1pq3 pq3
NOTE Remember that when
(c) 7a3b2 7a3b2 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) 9a 2b4 9a 2b4
Let’s look at some expressions involving more than two terms. The idea is just the same.
Example 5
© 2001 McGraw-Hill Companies
Combining Like Terms
NOTE The distributive property
Combine like terms.
can be used over any number of
like terms.
(a) 5ab 2ab 3ab
5ab (2ab) 3ab
(5 (2) 3)ab 6ab
*When an example requires simplification of an expression, that expression will be screened. The simplification
will then follow the equals sign.
118
CHAPTER 1
THE LANGUAGE OF ALGEBRA
Only like terms can be combined.
(b) 8x 2x
5y
(8 (2)) x 5y
6x
5y
Like terms
be writing out these steps, but
doing it mentally.
(c) 5m 8n
4m 3n
(5m 4m) (8n (3n))
9m
5n
Here we have used the associative
and commutative properties.
(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
As you have seen in arithmetic, subtraction can be performed directly. As this is the form
used for most of mathematics, we will use that form throughout this text. Just remember,
by using negative numbers, you can always rewrite a subtraction problem as an addition
problem.
Example 6
Combining Like Terms
Combine the like terms.
(a) 2xy 3xy 5xy
(b) 5a 2b 7b 8a
(2 3 5)xy
(5a 8a) (2b 7b)
4xy
3a 5b
CHECK YOURSELF 6
Combine like terms.
(a) 4ab 5ab 3ab 7ab
(b) 2x 7y 8x y
CHECK YOURSELF ANSWERS
1. (a) 2b4; (b) 5m, 3n; (c) 2s2, 3t, 6
2. (a) 8; (b) 5; (c) 1
2
2
2
3. The like terms are 5a b, a b, and 7a b
4. (a) 14b; (b) 9x2; (c) xy3; (d) 0
2
5. (a) 9m ; (b) 4ab 3a; (c) 9p 4q
6. (a) ab; (b) 6x 8y
© 2001 McGraw-Hill Companies
NOTE With practice you won’t
Like terms
Name
Exercises
1.6
Section
Date
List the terms of the following expressions.
1. 5a 2
2. 7a 4b
ANSWERS
1.
3. 4x3
4. 3x2
5. 3x2 3x 7
6. 2a 3 a2 a
2.
3.
4.
Circle the like terms in the following groups of terms.
5.
6.
7. 5ab, 3b, 3a, 4ab
7.
2
2
8. 9m , 8mn, 5m , 7m
8.
9.
9. 4xy2, 2x2y, 5x2, 3x2y, 5y, 6x2y
10.
11.
10. 8a2b, 4a2, 3ab2, 5a2b, 3ab, 5a2b
12.
Combine the like terms.
13.
11. 3m 7m
12. 6a2 8a2
13. 7b3 10b3
14. 7rs 13rs
14.
15.
16.
15. 21xyz 7xyz
16. 4mn2 15mn2
17.
18.
17. 9z2 3z2
18. 7m 6m
19. 5a3 5a3
20. 13xy 9xy
21. 19n2 18n2
22. 7cd 7cd
19.
© 2001 McGraw-Hill Companies
20.
21.
22.
23.
23. 21p2q 6p2q
24. 17r3s2 8r3s2
24.
25.
25. 10x 7x 3x
2
2
2
26. 13uv 5uv 12uv
26.
119
ANSWERS
27.
27. 9a 7a 4b
28. 5m2 3m 6m2
29. 7x 5y 4x 4y
30. 6a2 11a 7a2 9a
31. 4a 7b 3 2a 3b 2
32. 5p2 2p 8 4p2 5p 6
28.
29.
30.
31.
32.
33.
33.
2
4
m3 m
3
3
34.
1
4
a2 a
5
5
35.
13
3
x2 x5
5
5
36.
17
7
y7
y3
12
12
34.
35.
36.
37. 2.3a 7 4.7a 3
37.
38. 5.8m 4 2.8m 11
38.
Perform the indicated operations.
39.
39. Find the sum of 5a4 and 8a4.
40. Find the sum of 9p2 and 12p2.
41. Subtract 12a3 from 15a3.
42. Subtract 5m3 from 18m3.
40.
41.
42.
43. Subtract 4x from the sum of 8x and 3x.
43.
44.
44. Subtract 8ab from the sum of 7ab and 5ab.
45.
45. Subtract 3mn2 from the sum of 9mn2 and 5mn2.
46.
47.
46. Subtract 4x2y from the sum of 6x2y and 12x2y.
48.
Use the distributive property to remove the parentheses in each expression. Then simplify
by combining like terms.
49.
51.
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
52.
120
© 2001 McGraw-Hill Companies
50.
ANSWERS
53. Write a paragraph explaining the difference between n2 and 2n.
53.
54. Complete the explanation: “x3 and 3x are not the same because . . .”
54.
55. Complete the statement: “x 2 and 2x are different because . . .”
55.
56. Write an English phrase for each algebraic expression below:
56.
(a) 2x3 5x
(b) (2x 5)3
(c) 6(n 4)2
57. Work with another student to complete this exercise. Place , , or in the blank in
these statements.
12____21
57.
58.
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.
58. Work with other students on this exercise.
n2 1
n2 1
, n,
using odd values of n:
2
2
1, 3, 5, 7, etc. Make a chart like the one below and complete it.
Part 1: Evaluate the three expressions
© 2001 McGraw-Hill Companies
n
a
n2 1
2
bn
c
n2 1
2
a2
b2
c2
1
3
5
7
9
11
13
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.
121
ANSWERS
a.
Getting Ready for Section 1.7 [Section 0.3]
b.
Write the following using exponential notation.
c.
d.
(a) 4 4 4
(b) 6 6 6 6 6 6
e.
(c) 3 3 3 3 3
(d) (2) (2) (2)
f.
(e) (8) (8) (8) (8)
(f) 9 9 9 9 9 9 9 9
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.
e. (8)4
57.
a. 43
b. 66
c. 35
f. 98
© 2001 McGraw-Hill Companies
d. (2)3
55.
122