Download Combining Like Terms

Math 64
3.1 "Simplifying Algebraic Expressions"
Objectives:
*
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.
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,
Note:
:
If two variables or a number and a variable are next to each other, the indicated operation is multiplication.
For example,
:
The meaning of an exponent remains the same when the base is a variable.
For example,
:
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Combining Like Terms
The addends of an algebraic expression are called the terms of the expression.
For example,
:
* A term that is only a number has a special name. It is called a constant term, or simply a constant.
For example,
:
* The number factor of a variable is called the numerical coe¢ cient, or simply coe¢ cient.
For example,
:
* If no number is written next to a variable, then the coe¢ cient is understood to be 1.
For example,
:
* If a negative sign ( ) is written next to a variable, then the coe¢ cient is understood to be
For example,
1.
:
Example 1: (Finding the terms of the addends of an algebraic expression)
Identify the term and the coe¢ cient of the following expressions:
a)
4x3
Term
b) x4 + 7x2
Term
5x + 3x
Coe¢ cient
x3 + 5x2
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
Page: 1
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
3.1
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,
Example 2: (Combining like terms)
Simplify by combining like terms whenever possible.
a) 7x + 9x
b) 6x
c) 3x2 + x2 + 2b
5b
7a
d) 5y + 9y
a
y + 15
4
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
x
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
5)
Notes by Bibiana Lopez
Prealgebra by Elayn Martin-Gay
3.1
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
c) 3 (y + 4)
1) + 3
5 (y
b)
2) + 5
(4xy
d) 2 (3
x)
10) + 2 (3xy + 5)
5 (3
2x)
Finding Perimeter and Area
Example 6: (Application)
Find the perimeter of the following …gure.
Example 7: (Application)
Find the area of the following rectangle.
Page: 3
Notes by Bibiana Lopez