Simplifying Algebraic Expressions

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648 Chapter 8 An Introduction to Algebra

Objectives

S E C T I O N 8.2

1Simplify products.

2Use the distributive property.

3Identify like terms.

4Combine like terms.

EXAMPLE 7 Simplify by combining like terms, if possible:

a. 2x 9x b. 8p ( 2p) 4p c. 0.5s3 0.3s3 d. 4w 6 e.

a. 2x 9x b. 8p ( 2p) 4p c. 0.5s3 0.3s3 d. 4w 6 e.

In algebra, we frequently replace one algebraic expression with another that is equivalent and simpler in form. That process, called simplifying an algebraic expression, often involves the use of one or more properties of real numbers.

1 Simplify products.

The commutative and associative properties of multiplication can be used to simplify certain products. For example, let’s simplify 8(4x).

8(4x) 8 (4 x)	Rewrite 4x as 4 x.
(8 4) x	Use the associative property of multiplication to group 4 with 8.
32x	Do the multiplication within the parentheses.

We have found that 8(4x) 32x.We say that 8(4x) and 32x are equivalent expressions because for each value of x, they represent the same number. For example, if x 10, both expressions have a value of 320. If x 3, both expressions have a value of 96.

	If x 10			If x 3
8(4x) 8[4(10)]		32x 32(10)	8(4x) 8[4( 3)]		32x 32( 3)
8(40)		320	8( 12)		96
320			96




	same result			same result

Self Check 1

Simplify:

a.9 6s

b.4(6u)( 2)

c.23 32 m

d.36a29 yb

Now Try Problems 15,25,29,and 31

Success Tip By the commutative property of multiplication, we can change the order of factors. By the associative property of multiplication, we can change the grouping of factors.

	EXAMPLE 1 Simplify:
a. 9(3b) b. 15a(6) c. 3(7p)( 5) d.		8	3	r	e. 35a	4	xb
		3	8
					5

Strategy We will use the commutative and associative properties of multiplication to reorder and regroup the factors in each expression.

WHY We want to group all of the numerical factors of an expression together so that we can ﬁnd their product.

Solution

a. 9(3b) (9 3)b	Use the associative property of multiplication to
a. 9(3b) (9 3)b	regroup the factors.
	regroup the factors.
27b	Do the multiplication within the parentheses.
	Use the commutative property of multiplication to		3
			3
b. 15a(6) 15(6)a			15
b. 15a(6) 15(6)a	reorder the factors.		15
	reorder the factors.		6
			6
90a	Do the multiplication: 15(6) 90.		90
		Use the commutative and associative
c. 3(7p)( 5) [3(7)( 5)]p		properties of multiplication to reorder
		and regroup the factors.	21
[21( 5)]p		Multiply within the brackets.	21
[21( 5)]p		Multiply within the brackets.		5
				5
105p		Complete the multiplication within the brackets.	105

8.2 Simplifying Algebraic Expressions

649

d.83 38 r a83 38br

Use the associative property of multiplication to group the factors.

Multiply within the parentheses.

1 1

The product of a number and its reciprocal is 1: 83 83 1 .

e. 35a45xb a351 45bx

1 1

The coefﬁcient 1 need not be written.

Use the associative property of multiplication to regroup the factors.

a5 1 7 5 4bx Factor 35 as 5 7 and then remove the common factor 5.

	1
28x	Do the multiplication and then simplify:	28	28.
28x	Do the multiplication and then simplify:	28	28.
		1

2 Use the distributive property.

Another property that is often used to simplify algebraic expressions is the distributive property. To introduce it, we will evaluate 4(5 3) in two ways.

Method 1	Method 2
Use the order of operations:	Distribute the multiplication:
4(5 3) 4(8)	4(5 3) 4(5) 4(3)
32	20 12
	32

Each method gives a result of 32. This observation suggests the following property.

The Distributive Property

For any numbers a, b, and c,

a(b c) ab ac

The Language of Algebra To distribute means to give from one to several. You have probably distributed candy to children coming to your door on Halloween.

To illustrate one use of the distributive property, let’s consider the expression 5(x 3). Since we are not given the value of x, we cannot add x and 3 within the parentheses. However, we can distribute the multiplication by the factor of 5 that is outside the parentheses to x and to 3 and add those products.

5(x 3) 5(x) 5(3) Distribute the multiplication by 5.

5x 15

Do the multiplications.

The Language of Algebra Formally, it is called the distributive property of multiplication over addition. When we use it to write a product, such as 5(x 2), as a sum, 5x 10, we say that we have removed or cleared the parentheses.

650	Chapter 8 An Introduction to Algebra

Self Check 2

Multiply:

a.7(m 2)

b.80(8x 3)

c.24 ay6 38b

Now Try Problems 35, 37, and 39

EXAMPLE 2			x	9
Multiply: a. 8(m 9)	b. 12(4t 1)	c. 6a			b
Multiply: a. 8(m 9)	b. 12(4t 1)	c. 6a	3	2	b

Strategy In each case, we will distribute the multiplication by the factor outside the parentheses over each term within the parentheses.

WHY In each case, we cannot simplify the expression within the parentheses. To multiply, we must use the distributive property.

Solution

Self Check 3

Multiply:

a.5(2x 1)

b.9( y 4)

c.1(c 22)

Now Try Problems 43, 47, and 49

a.We read 8(m 9) as “eight times the quantity of m plus nine.” The word quantity alerts us to the grouping symbols in the expression.

		8(m 9) 8 m 8 9 Distribute the multiplication by 8.
					8m 72					Do the multiplications. Try to go directly to this step.
b. 12(4t 1) 12(4t) ( 12)(1) Distribute the multiplication by 12.											12
											12
					48t			( 12)		Do the multiplications.	4
					48t			( 12)		Do the multiplications.	48
									Write the result in simpler form. Recall that		48
					48t			12	Write the result in simpler form. Recall that
					48t			12	adding 12 is the same as subtracting 12.
									adding 12 is the same as subtracting 12.
	x		9			x		9
c. 6a				b	6		6			Distribute the multiplication by 6.
c. 6a	3		2	b	6	3	6	2		Distribute the multiplication by 6.
					1			1
					2 3 x			2 3 9 Factor 6 as 2 3 and then remove
					3				2	the common factors 3 and 2.
					3				2	the common factors 3 and 2.
					1				1
				2x 27

Since subtraction is the same as adding the opposite, the distributive property also holds for subtraction.

a(b c) ab ac

EXAMPLE 3

Multiply: a. 3(3b 4) b. 6( 3y 8) c. 1(t 9)

Strategy In each case, we will distribute the multiplication by the factor outside the parentheses over each term within the parentheses.

WHY In each case, we cannot simplify the expression within the parentheses. To multiply, we must use the distributive property.

Solution

a. 3(3b 4) 3(3b)	3(4)	Distribute the multiplication by 3.
9b 12		Do the multiplications. Try to go directly to this step.

Caution! A common mistake is to forget to distribute the multiplication over each of the terms within the parentheses.

3(3b 4) 9b 4

8.2 Simplifying Algebraic Expressions

651

b.6( 3y 8) 6( 3y) ( 6)(8)

18y ( 48)

18y 48

Distribute the multiplication by 6.

Do the multiplications.

Write the result in simpler form. Add the opposite of 48.

Another approach is to write the subtraction within the parentheses as addition of the opposite. Then we distribute the multiplication by 6 over the addition.

6( 3y 8) 6[ 3y ( 8)]	Add the opposite of 8.
6( 3y) ( 6)( 8)	Distribute the multiplication by 6.
18y 48	Do the multiplications.

c.1(t 9) 1(t) ( 1)(9)

t ( 9)

t 9

Distribute the multiplication by 1.

Do the multiplications.

Write the result in simpler form. Add the opposite of 9.

Notice that distributing the multiplication by 1 changes the sign of each term within the parentheses.

Caution! The distributive property does not apply to every expression that contains parentheses—only those where multiplication is distributed over addition (or subtraction). For example, to simplify 6(5x), we do not use the distributive property.

Correct	Incorrect
6(5x) (6 5)x 30x	6(5x) 30 6x 180x

The distributive property can be extended to several other useful forms. Since multiplication is commutative, we have:

(b c)a ba ca

For situations in which there are more than two terms within parentheses, we have:


	a(b c d) ab ac ad			a(b c d) ab ac ad
	EXAMPLE 4		Multiply:

a. (6x 4)		1	b. 2(a 3b)8	c. 0.3(3a 4b 7)

2

Strategy We will multiply each term within the parentheses by the factor (or factors) outside the parentheses.

WHY In each case, we cannot simplify the expression within the parentheses. To multiply, we use the distributive property.

Self Check 4

Multiply:

a.( 6x 24)13

b.6(c 2d)9

c.0.7(2r 5s 8)

Now Try Problems 53, 55, and 57

Solution

a. (6x 4)	1	(6x)	1	(4)	1	Distribute the multiplication by	1	.
	2		2		2
							2
		3x 2				Do the multiplications.

652	Chapter 8 An Introduction to Algebra
	b. This expression contains three factors.	Use the commutative property of
	2(a 3b)8 2 8(a 3b)	multiplication to reorder the factors.

16(a 3b) Multiply 2 and 8 to get 16.

16a 48b Distribute the multiplication by 16.

c.0.3(3a 4b 7) 0.3(3a) ( 0.3)(4b) ( 0.3)(7)

0.9a 1.2b 2.1 Do each multiplication.

0.30.3

3 4

0.91.2

0.3

7 2.1

We can use the distributive property to ﬁnd the opposite of a sum. For example, to ﬁnd (x 10), we interpret the symbol as a factor of 1, and proceed as follows:



(x 10) 1(x 10)		Replace the symbol with 1.
	1(x) ( 1)(10)	Distribute the multiplication by 1.
	x 10

In general, we have the following property.

Self Check 5

Simplify: ( 5x 18)

Now Try Problem 59

The Opposite of a Sum

The opposite of a sum is the sum of the opposites.

For any numbers a and b,

(a b) a ( b)

EXAMPLE 5 Simplify: ( 9s 3)

Strategy We will multiply each term within the parentheses by 1.

WHY The outside the parentheses represents a factor of 1 that is to be distributed.

Solution

( 9s 3) 1( 9s 3)	Replace the symbol in front
( 9s 3) 1( 9s 3)	of the parentheses with 1.
	of the parentheses with 1.
1( 9s) ( 1)(3)	Distribute the multiplication by 1.
9s 3	Do the multiplications. Try to go
	directly to this step.
	directly to this step.

3 Identify like terms.

Before we can discuss methods for simplifying algebraic expressions involving addition and subtraction, we need to introduce some new vocabulary.

Like Terms

Like terms are terms containing exactly the same variables raised to exactly the same powers. Any constant terms in an expression are considered to be like terms. Terms that are not like terms are called unlike terms.

								8.2 Simplifying Algebraic Expressions	653
Here are several examples.
	Like terms			Unlike terms
	4x and 7x				4x and 7y			The variables are not the same.
	10p2 and 25p2				10p and 25p2			Same variable, but different powers.
1		3	3	1		3	3
		c d and c d				c d and c		The variables are not the same.
	3	c d and c d			3	c d and c		The variables are not the same.

Success Tip When looking for like terms, don’t look at the coefﬁcients of the terms. Consider only the variable factors of each term. If two terms are like terms, only their coefﬁcients may differ.

	EXAMPLE 6	Identify the like terms in each expression:
a. 7r 5 3r		b. 6x4 6x2 6x c. 17m3 3 2 m3

Strategy First, we will identify the terms of the expression. Then we will look for terms that contain the same variables raised to exactly the same powers.

WHY If two terms contain the same variables raised to the same powers, they are like terms.

Self Check 6

Identify the like terms:

a.2x 2y 7y

b.5p2 12 17p2 2

Now Try Problem 63

Solution

a.7r 5 3r contains the like terms 7r and 3r.

b.Since the exponents on x are different, 6x4 6x2 6x contains no like terms.

c.17m3 3 2 m3 contains two pairs of like terms: 17m3 and m3 are like

terms, and the constant terms, 3 and 2, are like terms.

4 Combine like terms.

To add or subtract objects, they must be similar. For example, fractions that are to be added must have a common denominator.When adding decimals, we align columns to be sure to add tenths to tenths, hundredths to hundredths, and so on. The same is true when working with terms of an algebraic expression.They can be added or subtracted only if they are like terms.

This expression can be simpliﬁed	This expression cannot be simpliﬁed
because it contains like terms.	because its terms are not like terms.
3x 4x	3x 4y

Recall that the distributive property can be written in the following forms:

(b c)a ba ca (b c)a ba ca

We can use these forms of the distributive property in reverse to simplify a sum or difference of like terms. For example, we can simplify 3x 4x as follows:

3x 4x (3 4)x Use the form: ba ca (b c)a.

Success Tip Just as 3 apples plus 4 apples is 7 apples,

3x 4x 7x

654	Chapter 8 An Introduction to Algebra

We can simplify 15m2 9m2 in a similar way:

15m2 9m2 (15 9)m2 Use the form: ba ca (b c)a.

6m2

The Language of Algebra Simplifying a sum or difference of like terms is called combining like terms.

These examples suggest the following general rule.

Combining Like Terms

Like terms can be combined by adding or subtracting the coefﬁcients of the terms and keeping the same variables with the same exponents.

Self Check 7

Simplify, if possible:

a.3x 5x

b.6y ( 6y) 9y

c.4.4s4 3.9s4

d.4a 2

104 e. 7 c 7 c

Now Try Problems 67,71,79,and 83

Strategy We will use the distributive property in reverse to add (or subtract) the coefﬁcients of the like terms. We will keep the same variables raised to the same powers.

WHY To combine like terms means to add or subtract the like terms in an expression.

Solution

Self Check 8

Simplify:

a. 9h h	b. 9h h
c. 9h 8h	d. 8h 9h

Now Try Problems 73 and 77

a.Since 2x and 9x are like terms with the common variable x, we can combine them.

2x 9x 11x Think: (2 9)x 11x.

b. 8p ( 2p) 4p 6p Think: [ 8 ( 2) 4]p 6p. c. 0.5s3 0.3s3 0.2s3 Think: (0.5 0.3)s3 0.2s3.

d.Since 4w and 6 are not like terms, they cannot be combined. The expression 4w 6 doesn’t simplify.

e.		4	b	7	b	11	b	Think: 1	4	7	2b 11b.
	9			9		9
									9	9	9
	EXAMPLE 8							Simplify by combining like terms:

a.	16t 15t					b.		16t t		c.	15t 16t d. 16t t

Strategy As we combine like terms, we must be careful when working with terms such as t and t.

WHY Coefﬁcients of 1 and 1 are usually not written.

Solution

a.16t 15t t Think: (16 15)t 1t t.

b.16t t 15t Think: 16t 1t (16 1)t 15t.

c.15t 16t t Think: (15 16)t 1t t.

d.16t t 17t Think: 16t 1t (16 1)t 17t.

			8.2 Simplifying Algebraic Expressions				655
	EXAMPLE 9	Simplify: 6a2 54a	4a 36		Self Check 9
					Self Check 9
				Simplify: 7y2 21y 2y 6
Strategy First, we will identify any like terms in the expression. Then we will use				Simplify: 7y2 21y 2y 6
				Now Try		Problem 93
the distributive property in reverse to combine them.				Now Try		Problem 93
the distributive property in reverse to combine them.
WHY To simplify an expression we use properties of real numbers to write an
equivalent expression in simpler form.
Solution
We can combine the like terms that involve the variable a.
	6a2 54a 4a 36 6a2 50a 36		Think: (54 4)a 50a.
	6a2 54a 4a 36 6a2 50a 36		Think: (54 4)a 50a.

EXAMPLE 10 Simplify: 4(x 5) 5 (2x 4)

Strategy First, we will remove the parentheses. Then we will identify any like terms and combine them.

WHY To simplify an expression we use properties of real numbers, such as the distributive property, to write an equivalent expression in simpler form.

Solution

Here, the distributive property is used both forward (to remove parentheses) and in reverse (to combine like terms).

Replace the symbol

4(x 5) 5 (2x 4) 4(x 5) 5 1(2x 4) in front of (2x 4) with 1.

4x 20 5 2x 4

Distribute the multiplication by 4 and 1.

Think: (4 2)x 2x.

2x 19

Think: (20 5 4) 19.

ANSWERS TO SELF CHECKS
ANSWERS TO SELF CHECKS
1.	a. 54s	b.	48u	c. m		d. 8y 2. a.	7m 14	b. 640x 240 c. 4y 9
3.	a. 10x 5		b.	9y 36		c. c 22 4. a. 2x 8				b. 54c 108d
c.	1.4r 3.5s 5.6				5.	5x 18 6.	a. 2y and 7y		b.	5p2 and 17p2; 12 and 2
7.	a. 8x	b. 3y		c.	0.5s4 d. does not simplify			e. 76 c	8. a. 8h b. 10h c. h
d.	h 9.	7y2 19y 6				10. 21y 8

Self Check 10

Simplify:

6(3y 1) 2 ( 3y 4)

Now Try Problem 99


	S E C T I O N			8.2	STUDY SET
		VOCABULARY


	Fill in the blanks.
1.			To	the expression 5(6x) means to write it in
			simpler form: 5(6x) 30x.
2.			5(6x) and 30x are			expressions because for
			each value of x, they represent the same number.
3.			To perform the multiplication 2(x 8), we use the
				property.

4. We call (c 9) the

of a sum.

5.Terms such as 7x2 and 5x2, which have the same variables raised to exactly the same power, are called

terms.

6.When we write 9x x as 10x, we say we have

like terms.

656	Chapter 8 An Introduction to Algebra

CONCEPTS

7.a. Fill in the blanks to simplify the expression. 4(9t) ()t t

b. What property did you use in part a?

8.a. Fill in the blanks to simplify the expression.

6y 2 y y

b.What property did you use in part a?

9.Fill in the blanks.

a.2(x 4) 2x 8

b.2(x 4) 2x 8

c.2(x 4) 2x 8

d.2( x 4) 2x 8

10.Fill in the blanks to combine like terms.

a.4m 6m ()m m

b.30n2 50n2 ()n2 n2

c.12 32d 15 32d

d.Like terms can be combined by adding or

	subtracting the			of the terms and
	keeping the same			with the same
	exponents.
11. Simplify each expression, if possible.
a.	5(2x)	b.	5 2x
c.	6( 7x)	d. 6 7x
e.	2(3x)(3)	f.	2 3x 3

12. Fill in the blanks: Distributing multiplication by 1

changes the	of each term within the parentheses.
(x 10)		(x 10) x	10

NOTATION

13.Translate to symbols.

a.Six times the quantity of h minus four.

b.The opposite of the sum of z and sixteen.

14.Write an equivalent expression for the given expression using fewer symbols.

a. 1x	b. 1d	c. 0m
d. 5x	( 1)	e. 16t ( 6)

GUIDED PRACTICE

Simplify. See Example 1.
15.	3 4t											16.	9 3s
17.	9(7m)											18.	12n(8)
19.	5( 7q)											20.	7(5t)
21.	5t 60											22.	70a 10
23.	( 5.6x)( 2)											24.	( 4.4x)( 3)
25.	5(4c)(3)											26.	9(2h)(2)
27.	4( 6)( 4m)											28.	5( 9)( 4n)
29.	5	3					g					30.	9	7			k

	3			5									7		9
		5														4
31.	12a							xb				32.	15a						wb
31.	12a				12			xb				32.	15a			15			wb
		3														2
33.	8a					yb						34.	27a					xb
33.	8a		4			yb						34.	27a			3		xb
Multiply. See Example 2.
35.	5(x 3)											36.	4(x 2)
37.	3(4x 9)											38.	5(8x 9)
39.	45a				x				2			40.	35a			y				8
											b										b
					5				9		b					5				7	b
41.	0.4(x 4)											42.	2.2(2q 1)
Multiply. See Example 3.
43.	6(6c 7)											44.	9(9d 3)
45.	6(13c 3)											46.	2(10s 11)
47.	15( 2t 6)											48.	20( 4z 5)
49.	1( 4a 1)											50.	1( 2x 3)
Multiply. See Example 4.
51.	(3t 2)8											52.	(2q 1)9
53.	(3w 6)									2		54.	(2y 8)							1
53.	(3w 6)											54.	(2y 8)							2
									3											2
55.	4(7y 4)2											56.	8(2a 3)4
57.	25(2a 3b 1)											58.	5(9s 12t 3)
Simplify. See Example 5.
59.	(x 7)											60.	(y 1)
61.	( 5.6y 7)											62.	( 4.8a 3)

Identify the like terms in each expression, if any.

See Example 6.

63.3x 2 2x

64.3y 4 11y 6

65.12m4 3m3 2m2 m3

66.6x3 3x2 6x

Simplify by combining like terms. See Examples 7 and 8.


67.	3x 7x								68.	12y 15y
69.	4x 4x								70.	16y 16y
71.	7b2 27b2								72.	2c3 12c3
73.	13r 12r								74.	25s s
75.	36y y 9y								76.	32a a 5a
77.	43s3 44s3								78.	8j3 9j3
79.	9.8c 6.2c								80.	5.7m 4.3m
81.	0.2r ( 0.6r)								82.	1.1m ( 2.4m)
83.	3	t			1	t			84.	3		x		5			x
	5				5					16					16
	7				3					5				7
85.				x				x	86.				x					x
			16				16				18					18

Simplify by combining like terms, if possible. See Example 9.

87.	15y 10 y 20y	88.	9z 7 z 19z
89.	3x 4 5x 1	90.	4b 9 9b 9
91.	6m2 6m 6	92.	9a2 9a 9
93.	4x2 5x 8x 9	94.	10y2 8y y 7

Simplify. See Example 10.

95.2z 5(z 3)

96.12(m 11) 11

97.2(s2 7) (s2 2)

98.4(d2 3) (d2 1)

99.9(3r 9) 7(2r 7)

100.6(3t 6) 3(11t 3)

101.36a29 x 34b 36a12b

102.40a38 y 14b 40a45b

TRY IT YOURSELF

Simplify each expression.

103. 6 4( 3c 7)

105. 4r 7r 2r r

107. 24a 56 rb

109. a a a

111. 60a203 r 154 b

113. 5( 1.2x)

104. 10 5( 5g 1)

106. v 3v 6v 2v

3 1

108.4 2 g

110. t t t t

112. 72a78ƒ 89b

114. 5( 6.4c)

		8.2 Simplifying Algebraic Expressions	657
115.		(c 7) 2(c 3)
116.		(z 2) 5(3 z)
117.		a3 2a2 4a 2a2 4a 8
118.		c3 3c2 9c 3c2 9c 27
	APPLICATIONS

In Exercises 119–122, recall that the perimeter of a ﬁgure is
equal to the sum of the lengths of its sides.
119.		THE RED CROSS In 1891, Clara

		Barton founded the Red Cross. Its

symbol is a white ﬂag bearing a red	x
symbol is a white ﬂag bearing a red
cross. If each side of the cross has length

x, write an expression that represents the perimeter of the cross.

120.BILLIARDS Billiard tables vary in size, but all tables are twice as long as they are wide.

a.If the billiard table is x feet wide, write an expression that represents its length.

b.Write an expression that represents the perimeter of the table.

121.PING-PONG Write an expression that represents the perimeter of the Ping-Pong table.

x ft

		ft
		4)
x	ft	+
x		(
		x

122. SEWING Write

an expression that	(2x	–
represents the		15)	cm
			cm

length of the yellow	x cm
trim needed to			cm
outline a pennant	x–		cm
	(2	15)
	(2

with the given side lengths.

WRITING

123.Explain why the distributive property applies to 2(3 x) but not to 2(3x).

124.Explain how to combine like terms. Give an example.

REVIEW

Evaluate each expression for x 3, y 5, and z 0.

	x y2		2y 1
125.		126.		x
	2y 1 x		x

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