Learning Objectives
In Section R.2 you will review how to:
A. Identify terms, coeffi-
cients, and expressions
B. Create mathematical
models
C. Evaluate algebraic
expressions
D. Identify and use proper-
ties of real numbers
E. Simplify algebraic
expressions
R.2 Algebraic Expressions and the Properties of Real Numbers
To effectively use mathematics as a problem-solving tool, you must develop the ability
to translate written or verbal information into a mathematical model. After obtaining
a model, many applications require that you work effectively with algebraic terms and
expressions. The basic ideas involved are reviewed here.
A. Terms, Coefficients, and Algebraic Expressions
An algebraic term is a collection of factors that may include numbers, variables, or
expressions within parentheses. Here are some examples:
(1) 3 (2) (3) (4) (5) n (6)
If a term consists of a single nonvariable number, it is called a constant term. In (1),
3 is a constant term. Any term that contains a variable is called a variable term. We
call the constant factor of a term the numerical coefficient or simply the coefficient.
The coefficients for (1), (2), (3), and (4) are 3, 5, and respectively. In (5), the
coefficient of n is 1, since The term in (6) has two factors as written,
2 and The coefficient is 2.
An algebraic expression can be a single term or a sum or difference of terms. To
avoid confusion when identifying the coefficient of each term, the expression can be
rewritten using algebraic addition if desired: To identify the
coefficient of a rational term, it sometimes helps to decompose the term, rewriting it
using a unit fraction as in and
EXAMPLE 1
Identifying Terms and Coefficients
State the number of terms in each expression as given, then identify the coefficient
of each term.
a. b. c. d.
Rewritten: a. b. c. d.
Number of terms: two two one three
Coefficient(s): 2 and and and 5
Now try Exercises 7 through 14
B. Translating Written or Verbal Information into
a Mathematical Model
The key to solving many applied problems is finding an algebraic expression that accu-
rately models the situation. First, we assign a variable to represent an unknown quan-
tity, then build related expressions using words from the English language that suggest
a mathematical operation.
As mentioned earlier, variables that remind us of what they represent are often
used in the modeling process. Capital letters are also used due to their widespread
appearance in other fields.
EXAMPLE 2
Translating English Phrases into Algebraic Expressions
Assign a variable to the unknown number, then translate each phrase into an
algebraic expression.
a. twice a number, increased by five
b. six less than three times the width
2, 1,12
1
7
5
2x
2
11x2 511x 122
1
7
1x 32 12x22x 15y2
2x
2
x 51x 122
x 3
7
2x2x 5y
x
2
1
2
x.
n 2
5
1
5
1n 22
A B A 1B2.
1x 32.
1
#
n 1n n.
8,6,
21x 328n
2
5xy6P
A. You’ve just reviewed
how to identify terms,
coefficients, and expressions
R-13 13
College Algebra—
Solution
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