Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems:
College Algebra
Section 1.2: The Arithmetic of Algebraic
Expressions
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Objectives
o Components and terminology of algebraic
expressions.
o The field properties and their use in algebra.
o The order of mathematical operations.
o Basic set operations and Venn diagrams.
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Components and Terminology of Algebraic
Expressions
o Algebraic Expressions are made of constants and variables
combined by mathematical operations.
o Constants are fixed numbers.
o Variables are unspecified numbers.
o To evaluate an expression means to replace the variables with
constants, perform the mathematical operations and simplify.
o Terms are the parts of an algebraic expression joined by
addition (or subtraction).
o Factors of a term are the parts of a term that are joined by
multiplication (or division).
o The coefficient of a term is the constant factor of the term
while the remaining part of the term constitutes the variable
factor.
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Example 1: Components and Terminology
of Algebraic Expressions
Consider the algebraic expression
13 x 4 ( x 3  5 y )  7 x  19( x  y )
What are its terms?
4
3
13 x ( x  5 y), 7 x , 19( x  y )
What is the coefficient of each of the terms?
13, 7, 19
What is the variable factor of each of the terms?
x 4 ( x 3  5 y), x , ( x  y )
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Example 2: Components and Terminology
of Algebraic Expressions
Evaluate the following algebraic expression:
2 x 2  5( x  y) for x  1 and y  4
2 x 2  5( x  y)
 2(1)2  5  (1)  (4) 
Replace x with  1 and y with 4.
 2(1)  5(5)
Simplify.
 2  25
 27
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The Field Properties and Their Use in Algebra
The set of real numbers forms what is known mathematically as
a field and the properties below are called field properties. a, b
and c represent arbitrary real numbers.
Name of Property
Additive Version
Closure
a  b is real number
ab is a real number
Commutative
ab  ba
ab  ba
Associative
a  (b  c)  (a  b)  c
a(bc)  (ab)c
Identity
a  0  0 a  a
Inverse
Distributive
Multiplicative Version
a  1  1 a  a
1
a   1 (for a  0)
a  ( a )  0
a
a(b  c)  ab  ac
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Example 3: The Field Properties and Their
Use in Algebra
is the
a. 2  4  (1)  This
 2  4   2  (1) 
same as
 8   2
 2  3
6
6
This is the
b. 3  2
2  3
same as
 3  (2)
1
1

1  3
3
c. ( x  2 y )  3
, x  2y  0

x  2y 

x3  2 y
 3
1
x  2y
These two algebraic
expressions are the same
because of the distributive
property.
These two algebraic
expressions are the same
because of the commutative
property of addition.
Any non-zero expression,
when multiplied by its
reciprocal, yields the
multiplicative identity, 1.
1
Here, x 3  2 y is the
multiplicative inverse
of x 3  2 y.
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The Field Properties and Their Use in Algebra
Cancellation Properties
Throughout this table, A, B and C represent algebraic
expressions. The symbol  can be read as “if and only if” or “is
equivalent to.”
Property
Description
A  B  A C  B  C
For C  0,
A  B  AC  B C
Additive Cancellation
Adding the same quantity to both sides of
an equation results in an equivalent
equation
Multiplicative Cancellation
Multiplying both sides of an equation by
the same non-zero quantity results in an
equivalent equation
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The Field Properties and Their Use in Algebra
Zero-Factor Property
Let A and B represent algebraic expressions. If the
product of A and B is 0, then at least one of A and B is
itself 0. Using the symbol  for “implies,” we write
AB  0  A  0 or B  0
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Example 4: The Field Properties and Their
Use in Algebra
Verify the equivalence of the following algebraic expressions.
Remember that means “is
a. 5 x 2 y  2 z  z 3  2z  5 x 2 y  z 3 equivalent to.” Isthe equivalence
5 x 2 y  2z  z 3  2z
true?
Use additive cancellation by adding
2z
2z
 2z to both sides of the equation.
5x 2 y  z 3 5x 2 y  z 3
2
2
y
y


b. 5  x    10   x    2
4
4


2
1 
1
y
 5  x    10 
5 
5
4
2
2
y
y


 x    2  x    2
4
4


We can see that the equivalence is
true.
Is the equivalence true?
Use multiplicative cancellation by
multiplying both sides of the
1
equation by .
5
We can see that the equivalence is
true.
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Example 5: The Field Properties and
Their Use in Algebra
Verify the following algebraic expressions.
c. xy  14  0  0 Is the equivalence true?
No. While multiplying both sides of the equation by 0
0  xy  14  0
does result in 0=0, the two expressions are NOT
0  0  0  0 equivalent. Recall that the multiplicative cancellation
property requires that the number multiplied by both
sides is non-zero.
d.  x  y  x  y   0   x  y   0 or  x  y   0
Remember that the symbol  means “implies.” Is the above implication true?
Yes. Recall the zero-factor property. It states that if the product of two terms is
0, then at least one of the terms must be 0. Thus, x – y = 0 or x + y = 0. In
mathematics, “or” is not exclusive, so it could be true that both factors equal 0.
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The Order of Mathematical Operations
Order of Operations
1. If the expression is a fraction, simplify the numerator and
denominator individually, according to the guidelines in the
following steps.
2. Parentheses, braces and brackets are all used as grouping
symbols. Simplify expressions within each set of grouping
symbols, if any are present, working from the innermost
outward.
3. Simplify all powers (exponents) and roots.
4. Perform all multiplications and divisions in the expression in
the order they occur, working from left to right.
5. Perform all additions and subtractions in the expression in the
order they occur, working from left to right.
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Example 6: The Order of Mathematical
Operations
Follow the order of operations to solve the following.
14
The first step is to simplify the fraction.
a.12 
7
Next, we subtract.
 12  2
 10
b. 22
First we must simplify the power, 22.
Notice that this can be written as 1 times 22. Simplify the
2
 1 2
power.
Multiply.
 1 4
2
 4
Note: What if the expression was  2  ?
Then we would square 2 instead of 2 and the answer
would be 4.
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Example 7: The Order of Mathematical
Operations
Follow the order of operations to solve the following:
First, we simplify the numerator and
2  8 
5  2 9  1   denominator separately. In the numerator,
 2  first simplify the roots and powers and then
simplify the fractions. In the denominator
10  5  1
simplify the multiplication.
5   2  3  1 4  In the numerator, simplify inside the

parenthesis first. In the denominator, subtract
10  (5)
5 from 10.
5  (5)(4)
In the numerator, remember to multiply

before subtracting.
5
5   20 

5
5


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Basic Set Operations and Venn Diagrams
o Set operations, union and intersection, combine two
or more sets.
o A Venn diagram is a pictorial representation of a set
or sets and its aim is to indicate, through shading,
the outcome of set operations such as union and
intersection.
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Basic Set Operations and Venn Diagrams
A and B denote two sets and are represented in the Venn
diagram by circles. The operation of union is demonstrated
by shading. The symbol  is read “is an element of”.
The union of A and B,
denoted A  B, is the set
x | x  A or x  B.
That is, an element x is in
A
B
A  B, if it is in the set A,
the set B, or both. Note:
the union of A and B
contains both individual
sets.
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Basic Set Operations and Venn Diagrams
A and B denote two sets and are represented in the Venn
diagram by circles. The operation of intersection is
demonstrated by shading. The symbol  is read “is an
element of”.
The intersection of A and B,
denoted A  B, is the set
 x | x  A and x  B. That
A
B
is, an element x is in A  B,
if it is in both A and B.
Note: the intersection of A
and B is contained in each
individual set.
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Example 8: Basic Set Operations and Venn
Diagrams
Simplify each of the following set expressions.
a.  3,7    1,11 Since these two intervals overlap, their union is
best described with a single interval.
  3,11
b.  3,7    1,11 This intersection of two intervals can also be
described with a single interval.
  1,7 
These two intervals have no elements in
c.  2,5    5,7 
common so their intersection is the empty set.

d.  ,2   17,   The union of these two intervals constitutes the
entire set of real numbers.
  ,  

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