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The Distributive Property
Goals p Use the distributive property.
p Simplify expressions by combining like terms.
VOCABULARY
Distributive property
a(b c) ab ac
(b c)a ba ca
a(b c) ab ac
(b c)a ba ca
Coefficient In a term that is the product of a number and a
variable, the number is the coefficient of the variable.
Like terms Like terms are terms in an expression that have
the same variable raised to the same power.
Constant terms Constant terms are terms with no variable
factors.
Simplified An expression is simplified if it has no symbols of
grouping and if all the like terms have been combined.
Example 1
Using an Area Model
Find the area of a rectangle whose width is 5 and whose length
is x 3.
Solution You can find the area in two ways.
Area of One Rectangle
Area of Two Rectangles
5
5
x3
x
3
Answer Because both ways produce the same area, the following
statement is true.
5( x 3 ) 5(x) 5(3)
Lesson 2.6 • Algebra 1 Notetaking Guide
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THE DISTRIBUTIVE PROPERTY
The product of a and (b c):
a(b c) ab ac
(b c)a ba ca
Example: 5(x 2) 5x 10
Example: (x 4)8 8x 32
The product of a and (b c):
a(b c) ab ac
(b c)a ba ca
Example: 4(x 7) 4x 28
Example: (x 5)9 9x 45
Example 2
Using the Distributive Property
a. 8(x 9) 8 ( x ) 8 ( 9 )
Distribute the 8.
8x 72
In Example 2(b)
and (c), remember
that a factor with a
negative sign must
multiply each term of
an expression.
Forgetting to distribute the negative sign
to each term is a
common error.
Simplify.
b. 5(x 6) 5 ( x ) ( 5 )( 6 )
5x ( 30)
Distribute the ⴚ5.
Simplify.
5x 30
c. x(y 2x) x ( y ) ( x )( 2x )
xy (2x2)
Distribute the ⴚx.
Simplify.
xy 2x2
Example 3
Mental Math Calculations
Grocery Shopping Cereal is on sale for $2.51 per box. Use the
distributive property and mental math to calculate the cost of three
boxes of the cereal.
Solution
If you think of $2.51 as $2.50 $ .01 , the mental math is easier.
3(2.51) 3(2.50 0.01 )
3 ( 2.50 ) 3 ( 0.01 )
Use the distributive property.
7.50 0.03
Find the products mentally.
7.53
Find the sum mentally.
Answer The total cost is $ 7.53 .
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Write 2.51 as a sum.
Algebra 1 Notetaking Guide • Chapter 2
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Example 4
Simplifying by Combining Like Terms
a. 10x 13x ( 10 13 )x
Use the distributive property.
23x
Add coefficients.
b. 2xy 5 8xy 3
2xy (5) 8xy 3
In Example 4(c)
and (d), remember
that a negative sign
preceding a term
means that the term
is multiplied by
negative one.
Rewrite as an addition
expression.
2xy 8xy (5) 3
Group like terms.
10xy 2
Combine like terms.
c. 9xy 8y xy
9xy 8y (xy)
Rewrite as an addition
expression.
9xy (xy) 8y
Group like terms.
8xy 8y
Combine like terms.
d. x (3x x2)
x ( 1 )(3x x2)
Rewrite as an addition
expression.
x [( 1 )( 3x ) ( 1 )( x2 )]
Distribute the 1 .
x ( 3x ) ( x2 )
Multiply.
4x ( x2 )
Combine like terms.
4x x2
Simplify.
Checkpoint Complete the following exercises.
1. Simplify the expression:
x(4 x).
4x x2
2. Simplify the expression:
2y3 5y 4y3.
6y3 5y
3. Explain how to use the distributive property to mentally find the
cost of 8 erasers at $.49 each.
Figure out 8 p 0.50 then subtract 0.08 from the product,
because 8 p 0.49 8(0.50 0.01) 8 p 0.50 8 p 0.01 $3.92.
Lesson 2.6 • Algebra 1 Notetaking Guide
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