Basic Mathematics for College Students

Use the distributive property.

Another property that is often used to simplify algebraic expressions is the
distributive property.To introduce it, we will evaluate in two ways.

Method 1 Method 2

Use the order of operations: Distribute the multiplication:

Each method gives a result of 32. This observation suggests the following property.

The Distributive Property

For any numbers , , and ,

The Language of Algebra To distributemeans to give from one to several.

You have probably distributedcandy to children coming to your door on

Halloween.

To illustrate one use of the distributive property, let’s consider the expression

. Since we are not given the value of , we cannot add and 3 within the
parentheses. However, we can distribute the multiplication by the factor of 5 that is
outside the parentheses to and to 3 and add those products.

Distribute the multiplication by 5.

Do the multiplications.

The Language of Algebra Formally, it is called thedistributive property of

multiplication over addition.When we use it to write a product, such as ,

as a sum, 5 x 10 , we say that we haveremovedorclearedthe parentheses.

5(x2)

 5 x 15

5 (x3) 5 (x) 5 (3)

x

5(x3) x x

a(bc)abac

ab c

 32

 32  20  12

4( 5  3 )4( 8 ) 4 (53) 4 (5) 4 (3)

4(53)

2

8.2 Simplifying Algebraic Expressions 649

d.

Multiply within the parentheses.

The product of a number and its reciprocal is 1:.

The coefficient 1 need not be written.

e.

Factor 35 as 57 and then remove the common factor 5.

 28 x Do the multiplication and then simplify:^281  28.

a

5

1

 7  4

1  5

1

bx

Use the associative property of multiplication

35 a to regroup the factors.

4

5

xba

35

1



4

5

bx

r

8

1

31 

3

1

81 ^1

 1 r

Use the associative property of multiplication

to group the factors.

8

3



3

8

ra

8

3



3

8

br