Download 3.1 - UCR Math Dept.

Math 64
3.1 "Simplifying Algebraic Expressions"
Bibiana Lopez
Riverside City College
June 2010
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Objectives:
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Use properties of numbers to combine like terms.
Use properties of numbers to multiply expressions.
Simplify expressions by multiplying and then combining like terms.
Find the perimeter and area of …gures.
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Preliminaries:
Recall from Section 1.8 that a combination of operations on letters
(variables) and numbers is called an algebraic expression or simply
an expression. For example, 2x, n + 7, 3y 5 x .
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Preliminaries:
Recall from Section 1.8 that a combination of operations on letters
(variables) and numbers is called an algebraic expression or simply
an expression. For example, 2x, n + 7, 3y 5 x .
If two variables or a number and a variable are next to each other, the
indicated operation is multiplication. For example,
xy means x y , 3w means 3 w .
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Preliminaries:
Recall from Section 1.8 that a combination of operations on letters
(variables) and numbers is called an algebraic expression or simply
an expression. For example, 2x, n + 7, 3y 5 x .
If two variables or a number and a variable are next to each other, the
indicated operation is multiplication. For example,
xy means x y , 3w means 3 w .
The meaning of an exponent remains the same when the base is a
variable. For example,
x 2 = x x, y 4 = y y y y , m2 n3 = m m n n n .
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Combining Like Terms
The addends of an algebraic expression are called the terms of the
expression. For example,
x + 3 (Two terms) and 3y 2 + ( 6y ) + 4 (Three terms) .
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Combining Like Terms
The addends of an algebraic expression are called the terms of the
expression. For example,
x + 3 (Two terms) and 3y 2 + ( 6y ) + 4 (Three terms) .
A term that is only a number has a special name. It is called a
constant term, or simply a constant. For example, 3 and 4
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Combining Like Terms
The addends of an algebraic expression are called the terms of the
expression. For example,
x + 3 (Two terms) and 3y 2 + ( 6y ) + 4 (Three terms) .
A term that is only a number has a special name. It is called a
constant term, or simply a constant. For example, 3 and 4
.
The number factor of a variable is called the numerical coe¢ cient,
or simply coe¢ cient. For example,
The coe¢ cient of 3x is 3 and the coe¢ cient of 6y is 6 .
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Combining Like Terms
If no number is written next to a variable, then the coe¢ cient is
understood to be 1. For example, x = 1 x and w = 1 w .
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Combining Like Terms
If no number is written next to a variable, then the coe¢ cient is
understood to be 1. For example, x = 1 x and w = 1 w .
If a negative sign ( ) is written next to a variable, then the
coe¢ cient is understood to be 1. For example,
x = 1 x and
y2 = 1 y2 .
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Combining Like Terms
Example 1: (Finding the terms of the addends of an algebraic expression)
Identify the term and the coe¢ cient of the following expressions:
a)
4x 3 5x + 3x
b) x 4 + 7x 2 x 3 + 5x 2
Term
Coe¢ cient
Term
Coe¢ cient
Like Terms (or Similar Terms) are terms that contain the same variables
(if any) raised to the same powers.
Like Terms
Unlike Terms
Example 1a
Example 1b
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Combining Like Terms
Our objective here is to learn to simplify polynomials that contain like
terms. This process is called combining like terms. To understand how
to combine like terms, we need the distributive property as it applies to
integers.
Distributive Property:
If a, b, and c are numbers, then
Also, ac bc = (a b ) c .
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ac + bc = (a + b ) c
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Combining Like Terms
Example 2: (Combining like terms)
Simplify by combining like terms whenever possible.
a) 7x + 9x
b) 6x 7a a
c) 3x 2 + x 2 + 2b
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d) 5y + 9y
5b
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y + 15
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Combining Like Terms
The commutative and associative properties of addition and
multiplication can also help us simplify expressions.
Example 3: (Combining like terms)
Simplify:
a) 8 x + 4x 2 9x
b)
7y + 2 2y 9x + 12
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x
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Multiplying Expressions
We can also use properties of numbers to multiply expressions such as
3 (2x ) . By the associative property of multiplication, we can write the
product 3 (2x ) as (3 2) x, which simpli…es to 6x.
Example 4: (Multiplying expressions)
Use the distributive property to multiply:
a) 8 (y + 2)
b)
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3 (7m
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Simplifying Expressions
Next we will simplify expressions by …rst using the distributive
property to multiply and then combining any like terms.
Example 5: (Simplifying Expressions)
Simplify:
a) 5y 2 (y 1) + 3
b)
(4xy 10) + 2 (3xy + 5)
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Simplifying Expressions
c) 3 (y + 4)
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5 (y
2) + 5
d) 2 (3
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x)
5 (3
2x )
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Finding the Perimeter
Example 6: (Application)
Find the perimeter of the following …gure.
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Finding the Area
Example 7: (Application)
Find the area of the following rectangle.
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