Download Section 1.8 Algebraic Expressions: Like Terms

Section 1.8 Algebraic Expressions: Like Terms
Objectives
In this section, you will learn to:
•
•
•
•
To successfully complete this section,
you need to understand:
Identify the components of algebraic
•
terms and expressions.
•
Evaluate an expression with given
•
replacement values.
Combine like terms with integer coefficients.
Combine like terms with fractional coefficients.
Adding integers (1.3)
Subtracting real numbers (1.4)
Multiplying real numbers (1.5)
INTRODUCTION: SOME DEFINITIONS OF ALGEBRA
A variable is a letter that represents a number. A variable, like an operation sign and the radical, is an
algebraic symbol and an important component of what makes up an algebraic expression.
An algebraic expression is a combination of numbers and letters connected by operations and
grouping symbols. At its simplest, an algebraic expression can be just a single variable, such as
y, or a single number, such as 3.
Here are some examples of algebraic expressions:
w+8
5p + 6m 2
3y – x2
x2 – 1
2x + 5
When showing multiplication with variables, it is common to exclude the multiplication symbol. For
example, 5x is an abbreviated form of 5 · x. Likewise, xy means x · y, and 6xy 2 is an abbreviation for
6 · x · y 2.
In an algebraic expression, a number is called a constant because its value never changes. For example,
in the expression x + 3, x is a variable and 3 is a constant.
When a number is a multiplier of a variable, or a variable expression, we call the number a coefficient.
For example, in the expression 7x, the coefficient of x is 7.
ALGEBRAIC TERMS
-6m
3 – 2p5 , and 5k .
However, for our purposes here, a term is defined as the product of a number (coefficient) and one or
more variables, each with its own exponent. A term can also be a constant with no variable factors.
In its fullest definition, a term can be any quantity, including (7x3 + 9y),
1.8 Algebraic Expressions: Like Terms
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For example, the term 8x3 includes a coefficient of 8, and it includes a variable, x, with an exponent, 3.
The exponent applies only to the x and not to the 8.
Every term with a variable has both a numerical coefficient and an exponent, even if we don’t see them.
Consider, for example, the simple term x. It appears that this term has no coefficient and no exponent.
Actually, it has both. In fact,
•
•
•
x is the same as 1x or 1.x, so the coefficient of x is 1;
x is also the same as x1, so the exponent of x is 1; and
x can even be thought of as 1x1.
These implied 1’s—or invisible 1’s, as they are sometime called—are actually very important numbers.
Occasionally, we must show the coefficient 1 or the exponent 1 to assist us in simplifying an expression.
In this text, the variable part of a term is referred to as the term’s variable structure. This is especially
important when a term has more than one variable. For example, 10x2y3 has a variable structure of x2y3.
The exponent of x is 2, and the exponent of y is 3. In other words, this term has two factors of x and
three factors of y. Unabbreviated, the term is 10 · x · x · y · y · y, or 10xxyyy.
Think About It 1
Is the variable structure of 10x2y3 the same as or different from the variable
structure of 4x3y2? Explain your answer.
Example 1:
Given the term, identify the coefficient, the variable(s), and the exponent of each
variable.
Term
10x2y3
-3a
m4
-y
7
Coefficient
10
-3
1
-1
7*
Variable(s)
x and y
a
m
y
None
Exponent(s)
2 and 3
1
4
1
None
* 7 is a term that is really a constant term; by itself, it is not a coefficient as defined above.
1.8 Algebraic Expressions: Like Terms
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You Try It 1
Given the term, identify the coefficient, the variable(s), and the exponent of each
variable. Use Example 1 as a guide.
Term
a)
b)
c)
d)
3y4
-4
-2c
-m2p5
e)
f)
d5
w
Coefficient
Variable(s)
Exponent(s)
ALGEBRAIC EXPRESSIONS
The terms in an algebraic expression are often separated by a plus sign or a minus sign, and this allows
us to distinguish one term from another.
The expression -2x3 + 9x + 5 has three terms, each separated by a plus sign: -2x3, 9x, and 5.
There are also many algebraic expressions that include subtraction as one of the operations, such as
3x2 – 8x – 2. Recall (1.4) that subtraction can be written as adding the opposite, so this expression
can be rewritten as 3x2 + (-8x) + (-2). Now that it is written in addition only, the individual terms
become clearer. The terms are: 3x2, -8x, and -2.
What we see is that the coefficient of the subtracted term takes on the minus sign as a negative. This
means that the sign in front of a term—the plus sign or minus sign—belongs to the term and determines
whether the term is positive or negative.
The sign in front of a term belongs to that term.
Here are two expressions with their terms separated:
7x 2 – 2x – 5
Expression:
Terms:
+ 7x 2
1.8 Algebraic Expressions: Like Terms
- 2x
Expression:
-5
Terms:
119
- 4x2 + x – 6
- 4x 2
+ 1x
-6
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Example 2:
Identify the terms in each algebraic expression.
You Try It 2
Expression
Terms (separated by commas)
a)
b)
c)
d)
7x2
3x2
-7w
3a5
Three terms:
Three terms:
Two terms:
Four terms:
7x2,
3x2,
-7w,
3a5,
e)
-15y
One term:
-15y
+ 6x + 1
– 5x – 2
+ 8
– 2a3 – 4a + 9
6x, 1
-5x, -2
8
-2a3, -4a , 9
Identify the terms in each algebraic expression. Use Example 2 as a guide.
Expression
Terms
Expression
a)
5x2 – x + 3
b)
c3 – 6c2 – 8c
c)
-2x – 6y
d)
-5x3 + 6x2 + x – 4
Terms
It is also important to recognize the individual terms in an algebraic expression, as shown in this next
example.
Example 3:
Answer each question based on the expression
x4 – 8x2 + 3x – 5.
a)
What is the second term?
b)
What is the constant term?
c)
What is the coefficient of the first term?
d)
What is the exponent of the third term?
Procedure:
Keep the sign in front of the term in mind when identifying its coefficient.
Answer:
a)
-8x2
The subtraction sign in front of the second term means the term’s coefficient is negative.
b)
-5
The subtraction sign in front of the constant term means the constant is negative.
c)
1
x4 is the same as 1x4, so the coefficient is 1.
d)
1
3x is the same as 3x1, so the exponent of this term is 1.
1.8 Algebraic Expressions: Like Terms
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You Try It 3
Answer each question based on the expression
Example 3 as a guide.
9x3 – x2 – 4x + 6. Use
a)
What is the third term?
b)
What is the coefficient of the second term?
d)
What is the constant term?
d)
What is the exponent of the first term?
VARIABLES AS NUMBERS
Variables represent numbers. Sometimes we know what number a variable represents, and sometimes
we don’t. Recall (1.7) that when we are given the numerical value of a variable, that number is called a
replacement value.
For example, in the expression 3x + 5, if the value of x is -4,
we create a numerical expression by replacing x with -4. We
then evaluate the numerical expression using the order of
operations, as shown at right.
3x + 5
= 3(-4) + 5
= -12 + 5
= -7
Replace x with -4.
Multiply.
Add.
Notice that when we replace x with -4, parentheses are placed around the -4 to indicate
that the whole value of x is -4. Putting parentheses around the replacement value is
especially important when the value is a negative number.
Recall (1.7) the distinction between (-5)2 and -52. With parentheses, the negative is included in the
squaring, so (-5)2 = (-5)(-5) = 25. However, without parentheses, the negative is not included in the
squaring: -52 = -1 · 52 = -1 · (5)(5) = -25.
For this reason, if we are to square x, we must square the whole value of x, whether it is positive or
negative. We do so by putting parentheses around the replacement value.
For example, in the expression x2 – 3, replace x with -4 and
evaluate using the order of operations, as shown at right.
x2 – 3
= (-4)2 – 3
= 16 – 3
= 13
Replace x with -4.
Apply the exponent.
Subtract.
Furthermore, if an expression contains more than one variable, such as 2W + 2L, then each variable
will have its own replacement value, as demonstrated in Example 4.
1.8 Algebraic Expressions: Like Terms
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Example 4:
Evaluate each expression using the given replacement values.
a)
w + y
Replace w with 8
and y with -5
b)
2xy
Replace x with 5
and y with -6
c)
a2 – c
Replace a with -3
and c with 7
d)
2W + 2L
Replace W with 5
and L with 8
Procedure:
For each, apply the order of operations.
Answer:
a)
8 + (-5)
b)
= 3
c)
You Try It 4
2(5)(-6)
= -60
(-3)2 – 7
d)
2(5) + 2(8)
= 9–7
= 10 + 16
= 2
= 26
Evaluate each expression using the given replacement values. Use Example 4 as a
guide.
a)
2wc
Replace w with -5
and c with 8.
b)
hb
2
Replace h with 4
and b with 12.
c)
(r + n)2
Replace r with 2
and n with 6.
d)
y+4
3k
Replace y with 26
and k with -2.
Sometimes a variable, such as w, will appear in more than one part of an expression. When this
happens, the replacement value for w is the same for each occurrence of w.
18
We see two w’s in the expression 5w + w . If the replacement value for w is 3, then we must replace
each w in the expression with 3 (refer to Example 5a).
1.8 Algebraic Expressions: Like Terms
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Example 5:
a)
Evaluate each expression using the given replacement value(s).
18
5w + w
b)
Replace w with 3
x 2 + 3x – 7
c)
Replace x with -2
Procedure:
Replace each variable with the indicated value.
Answer:
a)
You Try It 5
18
5(3) + 3
4x + xy
Replace x with -5 and y with 3
(-2)2 + 3(-2) – 7
b)
c)
4(-5) + (-5)(3)
= 15 + 6
= 4 + (-6) – 7
= -20 + (-15)
= 21
= -9
= -35
Evaluate each expression using the given replacement value(s). Use Example 5 as a
guide.
a)
y 2 + 5y
Replace y with 6.
b)
x2 – 7x + 1
Replace x with -5.
c)
mp + 5p
Replace p with -4
and m with 9.
d)
a + ar
Replace a with 5
and r with -8.
LIKE TERMS
Two things are like each other if they share the same characteristics. In algebra, two or more terms are
considered to be like terms if they have exactly the same variable structure, even if the coefficients are
different.
For example, 3x2 and 9x2 are like terms because they have exactly the same variable structure, x2.
Similarly, 6m4p3 and 2m4p3 are like terms because their variable structure, m4p3, is identical.
However, 4c and -7p are not like terms because the variables are different.
Furthermore, if two terms have the same single variable with different exponents, such as 3x2 and 8x4,
then their variable structures are different and they are not like terms.
1.8 Algebraic Expressions: Like Terms
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Because constant terms have no variable factors at all, they are like all other constant terms. In the
example below, carefully examine which terms are like and which are not like. See if you can determine
why some terms (within the same expression) are not like the other terms.
Example 6:
In each expression, identify the like terms.
Expression
Like terms
a)
b)
c)
d)
e)
7x – 6x + 1
5x – 2 + 9
3a2 + 2b3 + 4a2 + 9b3
-15y + 6y – 3y
4x + 6y – 7
f)
-2x3 + 6x2 + 9x
You Try It 6
One pair of like terms:
One pair of like terms:
Two pairs of like terms:
All three are like terms:
No like terms:
No like terms:
7x and -6x
-2 and 9
3a2 and 4a2, 2b3 and 9b3
-15y and 6y and -3y
(None)
(None)
In each expression, identify the like terms. Use Example 6 as a guide.
Expression
Like Terms
Expression
a)
5x – x + 3
b)
x2 – 6x – 8x2
c)
4 – y – y
d)
9x5 + 2x5 – 6x5
Like Terms
COMBINING LIKE TERMS
In algebra we are asked to simplify a great deal. An expression is simplified when it is written in the
shortest and least complex form. When working with like terms, an expression is said to be simplified
when the like terms are combined. To combine like terms means to find their sum (or difference,
depending on the operation between the terms).
We can combine like terms because their variable structure is the same: 4h + 3h = 7h.
Notice that the resulting term, 7h, is like the two original terms because it has the same variable
structure. In this way, the only change is the coefficient; to get the coefficient of the result, 7, we add
the coefficients of the like terms, 3 and 4.
1.8 Algebraic Expressions: Like Terms
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Why does combining like terms work this way? Recall (1.1) that multiplication is an abbreviation for
repeated addition. So, when we must simplify the sum, 4h + 3h, we can think of it in this way:
4h means h + h + h + h, and 3h means h + h + h.
Therefore,
is the same as
4h
h + h + h + h
+
3h
+
h + h + h
which is abbreviated as 7h.
This is also why we can’t combine 2y + 5x. Unabbreviated, 2y + 5x is
(y + y)
+ (x + x + x + x + x)
In this case, we can’t say that we have 7 of anything that is the same.
The combining of like terms is not restricted to addition; we can also subtract like terms. In this way it
is like eliminating terms. Consider cutting 2 feet off of a 5 foot board; we are left with 3 feet:
5 feet – 2 feet = 3 feet.
Or, five x’s reduced by two x’s leaves three x’s:
5x –
2x = 3x.
We can also change subtraction to adding the opposite and write this expression as 5x + (-2x) = 3x.
Likewise, we can combine like terms with negative coefficients, as demonstrated in these examples:
Dollars:
Chuck’s account has $5 in it, but
he writes a check for $8. What is
the balance of Chuck’s account?
Temperature: At midnight it was -12° outside;
by 10 AM, the temperature had
risen by 5°. What was the
temperature at 10 AM?
5 dollars + (-8 dollars) = -3 dollars
Just as:
5y + (-8y) = -3y
-12 degrees + 5 degrees = -7 degrees
Just as:
-12w + 5w = -7w
To combine like terms we must add or subtract their numerical coefficients.
The resulting term has the same variable structure as the original terms.
In general, axn + bxn = (a + b)xn
If a term has no visible coefficient, then the coefficient is 1 or -1.
1.8 Algebraic Expressions: Like Terms
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Example 7:
Simplify each expression by combining like terms, if possible.
a)
7x + 6x
b)
2h + 3h + 6h
c)
-2 + 9
d)
-3a2 + 4a2
e)
4p + p
f)
c2 – 7c
g)
-2k – (-6k)
h)
5xy2 – 6xy2
i)
-4x + 4x
Procedure:
If the terms are like terms, then find the sum of the coefficients. The resulting term
has the same variable structure as the original terms. If they are not like terms,
rewrite the expression without combining.
Answer:
a)
13x
Add the coefficients together: 7 + 6 = 13; the result is another like term, 13x.
b)
11h
Yes, we can combine three terms if they are all like terms.
c)
7
All constants are like each other and can be combined.
d)
1a2 = a2
It is not necessary to write the coefficient, 1.
e)
5p
We can write the coefficient of the second term, p, as 1: 4p + 1p = 5p.
f)
c2 – 7c
These are not like terms and cannot be combined.
g)
4k
Write this subtraction as adding the opposite: -2k – (-6k) = -2k + 6k = 4k.
h)
-1xy2 = -xy2
5xy2 – 6xy2 = 5xy2 + (-6xy2) = -1xy2 . Notice that it is not necessary to
write the coefficient, -1, but the negative sign must still be present.
i)
0x = 0
0x means 0 · x = 0.
Notice that when the coefficient is either 1 or -1, as in Examples 4d) and 4h), it is not necessary to write
the coefficient.
Think About It 2
You Try It 7
9x3 and 9x2 are not like terms. Why not?
Simplify each expression by combining like terms. If they are not like terms,
rewrite the expression without combining. Use Example 7 as a guide.
a)
3y + 9y
b)
-6c2 + 7c2
c)
-x – (-6x)
d)
6m2 – 6m2
e)
5p2 – 6p
f)
4w3 – 7w3
1.8 Algebraic Expressions: Like Terms
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WHEN THE COEFFICIENT IS A FRACTION
If a variable is in the numerator of a fraction, the coefficient of the variable is the whole fraction, not just
3x
3
the numerator. For example, in 4 , the coefficient of x is not 3, it is the whole fraction, 4 . In other
3x
3
words, 4 = 4 x.
What allows us to separate the variable factor in the numerator from the rest of the fraction is the reverse
of the rule for multiplying fractions:
3
3 x
3·x
3x
3x
3·x
3 x
3
3
Multiply: 4 · x = 4 · 1 = 4 · 1 = 4 . Turning this around, 4 = 4 · 1 = 4 · 1 = 4 · x, or 4 x.
ax
a
In general, c = c x
Example 8:
Rewrite each fraction to show the true coefficient of the term’s variable. State what
the coefficient is.
a)
9m
5
b)
-3w2
8
c)
y
- 10
d)
x
6
Procedure:
Separate the variable from the fraction.
Answer:
a)
9
5 m
9
The coefficient is 5 .
b)
-3 2
8 w
-3
The coefficient is 8 .
c)
-1
10 y
-1
The coefficient is 10 .
d)
1
6 x
1
The coefficient is 6 .
You Try It 8
a)
5p
6
Rewrite each fraction to show the true coefficient of the term’s variable. State what
the coefficient is. Use Example 8 as a guide.
b)
7k
- 10
c)
v2
2
d)
-x
12
If the coefficients of like terms are fractions, then we can combine them as long as the fractions have a
common denominator. In other words, the coefficients must be like fractions.
If the fractional coefficients don’t already have the same denominator, they must be built up to have a
common denominator before the terms can be combined. If the fractional coefficients add to an
improper fraction (completely simplified), then it is appropriate to leave it as improper and not change it
to a mixed number.
1.8 Algebraic Expressions: Like Terms
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Example 9:
Combine each pair of like terms. Simplify the result, if possible.
7y
12
a)
Procedure:
5c2
7c2
b) 18 – 18
+ 11y
12
c)
2
5
4x
x + 15
If necessary, separate the variable from its fractional coefficient. To combine, we
must have like fractions (same denominator) and like terms (same variable structure).
Answer:
7
12
a)
11
y + 12 y
b)
5
18
7
c2 – 18 c2
18
The fractions can be simplified
by a factor of 6.
5
-7
= 18 c2 + 18 c2
3
3
Do not write 2 as a mixed number.
-2
= 18 c2
= 12 y
= 2 y
The fraction can be simplified
by a factor of 2.
1
= - 9 c2
c)
=
2
5
3
3
4
3
The LCD for 5 and 15 is 15. Multiply the first fraction by 3 .
x + 15 x
4
2
· 5 x + 15 x
6
4
= 15 x + 15 x
10
= 15 x
Now combine the like fractions and the like terms.
The fraction can be simplified by a factor of 5.
2
= 3 x
You Try It 9
a)
4
9
8x
x + 9
Combine each pair of like terms. Simplify the result, if possible. Use Example 9 as a
guide.
b)
w
14
1.8 Algebraic Expressions: Like Terms
-9w
– 14
c)
128
11
1
- 8 y2 + 4 y2
d)
4b3
9
3
+ 5b6
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EXPRESSIONS WITH TWO PAIRS OF LIKE TERMS
Because the sign in front of a term belongs to the term, we can—using the Commutative Property—
move terms around within an expression, as long as the sign moves with each term.
For example, we can write the terms of the expression 5x + 9 – 2x in a different order so that
the x-terms are together. Similarly, we can rearrange the terms of the expression -4y + 8x –
3y so that the y-terms are together.
Expression:
Recognize
the terms:
5x + 9 – 2x
+ 5x
Rewrite the
expression:
+9
Expression:
Recognize
the terms:
- 2x
- 4y + 8x – 3y
- 4y
Rewrite the
expression:
5x – 2x + 9
+ 8x
- 3y
8x – 4y – 3y
If an expression has more than one pair of like terms, such as 3x2 + 1 + 5x2 – 6, we can gather each
pair separately and combine only those terms that are like each other.
3x2 + 1 + 5x2 – 6
=
3x2 + 5x2
=
8x2
=
8x2 – 5
Example 10:
a)
+ 1–6
+
Combine the like terms individually: 1 – 6 = 1 + (-6) = -5
This can be written as 8x2 – 5.

We cannot combine any further.
(-5)
Combine like terms.
5x – 2x + 2 + 8
Procedure:
Rearrange the terms so that like terms are together.
b)
2w2 + 3w + 7w2 – 5w
c)
3
4
1
y – 5 – 4 y + 2
First, rearrange the terms by gathering like terms together.
Second, combine any like terms.
Answer:
a)
=
5x – 2x + 2 + 8
The like terms are already gathered together.
Let’s underline them to see the like terms.
5x – 2x
Combine like terms.
+ 2+8
= 3x + 10
1.8 Algebraic Expressions: Like Terms
We can’t combine further.
129
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2w2 + 3w + 7w2 – 5w
b)
Rearrange the terms to gather the like terms in pairs.
= 2w2 + 7w2 + 3w – 5w
c)
=
9w2 + (-2w)
=
9w2 – 2w
3
4
Combine the like terms. (Underlining is not necessary.)
The final expression has two terms: 9w2 and -2w,
and we can write it as subtraction.
1
y – 5 – 4 y + 2
3
Rearrange the terms to gather the like terms in pairs.
1
We can write minus 5 as plus (-5). Also, the
fractions can combine directly.
= 4 y – 4 y – 5+ 2
2
The fraction can be simplified by a factor of 2.
Also, (-5) + 2 = -3.
= 4 y + (-5) + 2
1
= 2 y + (-3)
1
= 2 y
You Try It 10
We can write plus (-3) as minus 3.
– 3
Simplify each expression by combining like terms. Use Example 10 as a guide.
a)
y + 4y + 6y2 – 2y2
b)
3x2 – 8x + 2x – 4x2
c)
5
1
-2c2 + 12 + c2 – 12
d)
1
3
1.8 Algebraic Expressions: Like Terms
130
1
m – 1 + 6 m – 6
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Copying is prohibited.
You Try It Answers
You Try It 1:
Term
Coefficient
Variable
Exponent
a)
3y4
3
y
4
b)
-4
-4
None
None
c)
-2c
-2
c
1
d)
-m2p5
-1
m and p
2 and 5
e)
d5
1
d
5
f)
w
1
w
1
You Try It 2:
a)
d)
5x2, -x, 3
-5x3, 6x2, x, -4
c3, -6c2, -8c
You Try It 3:
a)
-4x
b)
-1
c)
6
d)
3
You Try It 4:
a)
-80
b)
24
c)
64
d)
-5
You Try It 5:
a)
66
b)
61
c)
-56
d)
-35
You Try It 6:
a)
c)
5x and -x
-y and -y
b)
d)
x2 and -8x2
9x5, 2x5, and -6x5
You Try It 7:
a)
d)
12y
0
You Try It 8:
a)
b)
1c2 or c2
5p2 – 6p
b)
e)
c)
5
5
6 p; the coefficient is 6 .
1
1 2
2 v ; the coefficient is 2 .
You Try It 9:
a)
4
3
You Try It 10:
a)
5y + 4y2
c)
1
-1c2 + 3
x
1.8 Algebraic Expressions: Like Terms
b)
5
7
c)
w
1
or -c2 + 3
131
c)
f)
-2x, -6y
5x
-3w3
d)
7
7
- 10 k; the coefficient is - 10 .
1
1
- 12 x; the coefficient is - 12 .
c)
- 8 y2
d)
b)
-1x2 + (-6x)
or -x2 – 6x
d)
1
2
b)
9
m + (-7)
23
18
b3
1
or 2 m – 7
© Robert H. Prior, 2014
Copying is prohibited.
Section 1.8 Exercises
Think Again.
1.
Why aren’t 9x3 and 9x2 like terms? (Refer to Think About It 1)
2.
What does “the sign in front of a term belongs to that term” mean?
3.
Can 5x represent a negative number? Explain your answer or show an example that supports
your answer.
4.
Can -6x represent a positive number? Explain your answer or show an example that supports
your answer.
Focus Exercises.
Given the term, identify the coefficient, the variable, and the exponent of the variable.
Term
Coefficient
Variable
Exponent
Term
5.
4m2
6.
-3y
7.
9
8.
y3
9.
-d5
10.
-n
11.
Answer each question based on the expression
Coefficient
Variable
Exponent
3x4 – 5x3 + x – 9.
a)
What is the first term?
b)
What is the second term?
c)
What is the coefficient of the third term?
d)
What is the constant term?
e)
What is the exponent of the first term?
f)
What is the exponent of the third term?
12.
Answer each question based on the expression
-8y + y2 – 3y4 + 1.
a)
What is the first term?
b)
What is the second term?
c)
What is the coefficient of the third term?
d)
What is the constant term?
e)
What is the exponent of the first term?
f)
What is the exponent of the third term?
1.8 Algebraic Expressions: Like Terms
132
© Robert H. Prior, 2014
Copying is prohibited.
Evaluate the following according to the replacement value given.
13.
3x + 4; replace x with 2.
14.
3v ÷ 9; replace v with 6.
15.
19 – 5x; replace x with 1.
16.
bc ÷ 2; replace b with 11 and c with 4.
17.
y 2 + 3y; replace y with 4.
18.
(P + 10) ÷ P; replace P with 5.
19.
5u – v; replace u with 9 and v with 20
20.
(p + r) ÷ (p – r); replace p with 11 and r with 9.
Simplify. Combine like terms, if possible.
21.
2x + 5x
22.
4x2 + 9x2
23.
4a – 5a
24.
-12x2 – 10x
25.
-9y2 – (-11y2)
26.
3m – 6m
27.
-8r + 8r
28.
-p2 + 9p2
29.
k + (-7k)
30.
v2 – 2v2
31.
2s – (-4m)
32.
-3v + 3v
33.
5y + 7y – 2y
34.
5x – 3x – 7x
35.
-8c + 4c + 10c
36.
-13x + 8x – 2x
37.
-5y2 – 6y2 + 3y2
38.
-7p – 6p – 15p
39.
7
13
- 20 y – 20 y
40.
4h
15
2h
+ 15
41.
1k
5k
- 6 + 9
42.
7p
1p
10 – 6
43.
17
11
- 30 b2 + 30 b2
44.
11x2
24
7x2
+ 24
Simplify by combining pairs of like terms.
45.
4x + 7x + 8 – 5
46.
6y + 5y + 7 – 6
47.
3x – 4x + 4 – 6
48.
5p – 6p + 3 – 9
49.
-3a + 7a + 4 – 9
50.
-5a + 6a + 2 – 7
1.8 Algebraic Expressions: Like Terms
133
© Robert H. Prior, 2014
Copying is prohibited.
51.
3w + 6w2 + 2w – w2
52.
4m2 + 5m + 3m2 – m
53.
-6 – 2x + 5 – x
54.
-x2 – 6x + 4x2 – x
55.
-5c – 3b + c – b
56.
-3x – 7y + x – y
Think Outside the Box.
Evaluate the following according to the replacement value given.
57.
Is it possible to replace x with 2 in the expression (6 – x) ÷ (6 – 3x)? Explain your answer.
58.
Is it possible to replace y with 4 in the expression (8 – 2x) ÷ (8 – x)? Explain your answer.
Find the algebraic expression, in terms of x, that represents the perimeter of the given rectangle.
59.
60.
10x + 7
4 –!6x
20 –!2x
3x + 7
61.
Referring to the rectangle in #107, does the diagram make sense if x = 1? Explain your answer.
62.
Referring to the rectangle in #108, does the diagram make sense if x = 1? Explain your answer.
1.8 Algebraic Expressions: Like Terms
134
© Robert H. Prior, 2014
Copying is prohibited.