Basic Mathematics for College Students

Caution! The distributive property does not apply to every expression that

contains parentheses—only those where multiplication is distributed over

addition (or subtraction). For example, to simplify , we do not use the

distributive property.

Correct Incorrect

The distributive property can be extended to several other useful forms. Since
multiplication is commutative, we have:

For situations in which there are more than two terms within parentheses, we have:

a(bcd)abacad a(bcd)abacad

(bc)abaca (bc)abaca

6(5x)(65)x 30 x 6(5x) 30  6 x 180 x

6(5x)

8.2 Simplifying Algebraic Expressions 651

b. Distribute the multiplication by.

Do the multiplications.

Another approach is to write the subtraction within the parentheses as addition of
the opposite. Then we distribute the multiplication by over the addition.

Add the opposite of 8.

Distribute the multiplication by.

Do the multiplications.

c. Distribute the multiplication by.

Do the multiplications.

Notice that distributing the multiplication by changes the signof each term
within the parentheses.

 1

Write the result in simpler form. Add the opposite

t (^9) of 9.
t(9)
 1 (t9) 1 (t)( 1 )(9)  1
 18 y 48
 6 ( 3 y)( 6 )(8)  6
 6 ( 3 y8) 6 [ 3 y(8)]

 6

Write the result in simpler form.

 18 y (^48) Add the opposite of 48.
 18 y(48)
 6 ( 3 y8) 6 ( 3 y)( 6 )(8)  6
EXAMPLE (^4) Multiply:
a. b. c.
StrategyWe will multiply each term within the parentheses by the factor (or
factors) outside the parentheses.
WHYIn each case, we cannot simplify the expression within the parentheses. To
multiply, we use the distributive property.
Solution
a. Distribute the multiplication by.
 3 x 2 Do the multiplications.
(6x4) 21

1

2

(6x)

1

2

(4)

1

2

(6x4) 2(a 3 b)8 0.3(3a 4 b7)

1

2

Self Check 4

Multiply:

a.

b.

c.

Now TryProblems 53, 55, and 57

0.7(2r 5 s8)

6(c 2 d)9

( 6 x24)

1

3