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Distributive Property Practice Problems

Hopefully you've read and understood the distributive property lesson. If you are ready, let's move on and practice our skill!

Directions: Use the distributive property to simplify each expression.

NOTE: The answers are down below the practice problems. Make sure you check your answers carefully!

1. 5(x+6) =

2. 3(2x+1) =

3. 2(2x-3) =

4. 6(x-4) =

5. 8(2x-3) =

6.    -3(x + 4)=

7.    -2(3x - 1)=

8.    -10(2x + 1/5 )=

9.    -18(x - 2/3)=

10.    1/2(x -4) =

1.  5x + 30

2.  6x +3

3.  4x - 6

4.  6x - 24

5.  16x - 24

6.  -3x - 12

7.  -6x +2

8.  -20x - 2

9.  -18x + 12

10.  1/2x - 2

Ok... The distributive property isn't so bad is it? If you're ready, you can move onto simplifying algebraic expressions!

If you didn't do so well on the distributive property, you may need to go back and review Adding Integers , Subtracting Integers , or Multiplying and Dividing Integers.

• Pre-Algebra
• Distributive Property
• Practice Problems

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7.3 Distributive Property

Learning objectives.

By the end of this section, you will be able to:

• Simplify expressions using the distributive property
• Evaluate expressions using the distributive property

Be Prepared 7.7

Before you get started, take this readiness quiz.

Multiply: 3 ( 0.25 ) . 3 ( 0.25 ) . If you missed this problem, review Example 5.15

Be Prepared 7.8

Simplify: 10 − ( −2 ) ( 3 ) . 10 − ( −2 ) ( 3 ) . If you missed this problem, review Example 3.51

Be Prepared 7.9

Combine like terms: 9 y + 17 + 3 y − 2 . 9 y + 17 + 3 y − 2 . If you missed this problem, review Example 2.22 .

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $9.25 ; $9.25 ; that is, 9 9 dollars and 1 1 quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

They need 3 3 times $9 , $9 , so $27 , $27 , and 3 3 times 1 1 quarter, so 75 75 cents. In total, they need $27.75 . $27.75 .

If you think about doing the math in this way, you are using the Distributive Property.

Distributive Property

If a , b , c a , b , c are real numbers, then

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression 3 ( x + 4 ) , 3 ( x + 4 ) , the order of operations says to work in the parentheses first. But we cannot add x x and 4 , 4 , since they are not like terms. So we use the Distributive Property, as shown in Example 7.17 .

Example 7.17

Simplify: 3 ( x + 4 ) . 3 ( x + 4 ) .

Try It 7.33

Simplify: 4 ( x + 2 ) . 4 ( x + 2 ) .

Try It 7.34

Simplify: 6 ( x + 7 ) . 6 ( x + 7 ) .

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 7.17 would look like this:

Example 7.18

Simplify: 6 ( 5 y + 1 ) . 6 ( 5 y + 1 ) .

Try It 7.35

Simplify: 9 ( 3 y + 8 ) . 9 ( 3 y + 8 ) .

Try It 7.36

Simplify: 5 ( 5 w + 9 ) . 5 ( 5 w + 9 ) .

The distributive property can be used to simplify expressions that look slightly different from a ( b + c ) . a ( b + c ) . Here are two other forms.

Other forms

Example 7.19

Simplify: 2 ( x − 3 ) . 2 ( x − 3 ) .

Try It 7.37

Simplify: 7 ( x − 6 ) . 7 ( x − 6 ) .

Try It 7.38

Simplify: 8 ( x − 5 ) . 8 ( x − 5 ) .

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Example 7.20

Simplify: 3 4 ( n + 12 ) . 3 4 ( n + 12 ) .

Try It 7.39

Simplify: 2 5 ( p + 10 ) . 2 5 ( p + 10 ) .

Try It 7.40

Simplify: 3 7 ( u + 21 ) . 3 7 ( u + 21 ) .

Example 7.21

Simplify: 8 ( 3 8 x + 1 4 ) . 8 ( 3 8 x + 1 4 ) .

Try It 7.41

Simplify: 6 ( 5 6 y + 1 2 ) . 6 ( 5 6 y + 1 2 ) .

Try It 7.42

Simplify: 12 ( 1 3 n + 3 4 ) . 12 ( 1 3 n + 3 4 ) .

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

Example 7.22

Simplify: 100 ( 0.3 + 0.25 q ) . 100 ( 0.3 + 0.25 q ) .

Try It 7.43

Simplify: 100 ( 0.7 + 0.15 p ) . 100 ( 0.7 + 0.15 p ) .

Try It 7.44

Simplify: 100 ( 0.04 + 0.35 d ) . 100 ( 0.04 + 0.35 d ) .

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

Example 7.23

Simplify: m ( n − 4 ) . m ( n − 4 ) .

Notice that we wrote m · 4 as 4 m . m · 4 as 4 m . We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

Try It 7.45

Simplify: r ( s − 2 ) . r ( s − 2 ) .

Try It 7.46

Simplify: y ( z − 8 ) . y ( z − 8 ) .

The next example will use the ‘backwards’ form of the Distributive Property, ( b + c ) a = b a + c a . ( b + c ) a = b a + c a .

Example 7.24

Simplify: ( x + 8 ) p . ( x + 8 ) p .

Try It 7.47

Simplify: ( x + 2 ) p . ( x + 2 ) p .

Try It 7.48

Simplify: ( y + 4 ) q . ( y + 4 ) q .

When you distribute a negative number, you need to be extra careful to get the signs correct.

Example 7.25

Simplify: −2 ( 4 y + 1 ) . −2 ( 4 y + 1 ) .

Try It 7.49

Simplify: −3 ( 6 m + 5 ) . −3 ( 6 m + 5 ) .

Try It 7.50

Simplify: −6 ( 8 n + 11 ) . −6 ( 8 n + 11 ) .

Example 7.26

Simplify: −11 ( 4 − 3 a ) . −11 ( 4 − 3 a ) .

You could also write the result as 33 a − 44 . 33 a − 44 . Do you know why?

Try It 7.51

Simplify: −5 ( 2 − 3 a ) . −5 ( 2 − 3 a ) .

Try It 7.52

Simplify: −7 ( 8 − 15 y ) . −7 ( 8 − 15 y ) .

In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, − a = −1 · a . − a = −1 · a .

Example 7.27

Simplify: − ( y + 5 ) . − ( y + 5 ) .

Try It 7.53

Simplify: − ( z − 11 ) . − ( z − 11 ) .

Try It 7.54

Simplify: − ( x − 4 ) . − ( x − 4 ) .

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Example 7.28

Simplify: 8 − 2 ( x + 3 ) . 8 − 2 ( x + 3 ) .

Try It 7.55

Simplify: 9 − 3 ( x + 2 ) . 9 − 3 ( x + 2 ) .

Try It 7.56

Simplify: 7 x − 5 ( x + 4 ) . 7 x − 5 ( x + 4 ) .

Example 7.29

Simplify: 4 ( x − 8 ) − ( x + 3 ) . 4 ( x − 8 ) − ( x + 3 ) .

Try It 7.57

Simplify: 6 ( x − 9 ) − ( x + 12 ) . 6 ( x − 9 ) − ( x + 12 ) .

Try It 7.58

Simplify: 8 ( x − 1 ) − ( x + 5 ) . 8 ( x − 1 ) − ( x + 5 ) .

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part ⓐ , we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example 7.30

When y = 10 y = 10 evaluate: ⓐ 6 ( 5 y + 1 ) 6 ( 5 y + 1 ) ⓑ 6 · 5 y + 6 · 1 . 6 · 5 y + 6 · 1 .

Notice, the answers are the same. When y = 10 , y = 10 ,

Try it yourself for a different value of y . y .

Try It 7.59

Evaluate when w = 3 : w = 3 : ⓐ 5 ( 5 w + 9 ) 5 ( 5 w + 9 ) ⓑ 5 · 5 w + 5 · 9 . 5 · 5 w + 5 · 9 .

Try It 7.60

Evaluate when y = 2 : y = 2 : ⓐ 9 ( 3 y + 8 ) 9 ( 3 y + 8 ) ⓑ 9 · 3 y + 9 · 8 . 9 · 3 y + 9 · 8 .

Example 7.31

When y = 3 , y = 3 , evaluate ⓐ −2 ( 4 y + 1 ) −2 ( 4 y + 1 ) ⓑ −2 · 4 y + ( −2 ) · 1 . −2 · 4 y + ( −2 ) · 1 .

Try It 7.61

Evaluate when n = −2 : n = −2 : ⓐ −6 ( 8 n + 11 ) −6 ( 8 n + 11 ) ⓑ −6 · 8 n + ( −6 ) · 11 . −6 · 8 n + ( −6 ) · 11 .

Try It 7.62

Evaluate when m = −1 : m = −1 : ⓐ −3 ( 6 m + 5 ) −3 ( 6 m + 5 ) ⓑ −3 · 6 m + ( −3 ) · 5 . −3 · 6 m + ( −3 ) · 5 .

Example 7.32

When y = 35 y = 35 evaluate ⓐ − ( y + 5 ) − ( y + 5 ) and ⓑ − y − 5 − y − 5 to show that − ( y + 5 ) = − y − 5 . − ( y + 5 ) = − y − 5 .

Try It 7.63

Evaluate when x = 36 : x = 36 : ⓐ − ( x − 4 ) − ( x − 4 ) ⓑ − x + 4 − x + 4 to show that − ( x − 4 ) = − x + 4 . − ( x − 4 ) = − x + 4 .

Try It 7.64

Evaluate when z = 55 : z = 55 : ⓐ − ( z − 10 ) − ( z − 10 ) ⓑ − z + 10 − z + 10 to show that − ( z − 10 ) = − z + 10 . − ( z − 10 ) = − z + 10 .

ACCESS ADDITIONAL ONLINE RESOURCES

• Model Distribution
• The Distributive Property

Section 7.3 Exercises

Practice makes perfect.

In the following exercises, simplify using the distributive property.

4 ( x + 8 ) 4 ( x + 8 )

3 ( a + 9 ) 3 ( a + 9 )

8 ( 4 y + 9 ) 8 ( 4 y + 9 )

9 ( 3 w + 7 ) 9 ( 3 w + 7 )

6 ( c − 13 ) 6 ( c − 13 )

7 ( y − 13 ) 7 ( y − 13 )

7 ( 3 p − 8 ) 7 ( 3 p − 8 )

5 ( 7 u − 4 ) 5 ( 7 u − 4 )

1 2 ( n + 8 ) 1 2 ( n + 8 )

1 3 ( u + 9 ) 1 3 ( u + 9 )

1 4 ( 3 q + 12 ) 1 4 ( 3 q + 12 )

1 5 ( 4 m + 20 ) 1 5 ( 4 m + 20 )

9 ( 5 9 y − 1 3 ) 9 ( 5 9 y − 1 3 )

10 ( 3 10 x − 2 5 ) 10 ( 3 10 x − 2 5 )

12 ( 1 4 + 2 3 r ) 12 ( 1 4 + 2 3 r )

12 ( 1 6 + 3 4 s ) 12 ( 1 6 + 3 4 s )

r ( s − 18 ) r ( s − 18 )

u ( v − 10 ) u ( v − 10 )

( y + 4 ) p ( y + 4 ) p

( a + 7 ) x ( a + 7 ) x

−2 ( y + 13 ) −2 ( y + 13 )

−3 ( a + 11 ) −3 ( a + 11 )

−7 ( 4 p + 1 ) −7 ( 4 p + 1 )

−9 ( 9 a + 4 ) −9 ( 9 a + 4 )

−3 ( x − 6 ) −3 ( x − 6 )

−4 ( q − 7 ) −4 ( q − 7 )

−9 ( 3 a − 7 ) −9 ( 3 a − 7 )

−6 ( 7 x − 8 ) −6 ( 7 x − 8 )

− ( r + 7 ) − ( r + 7 )

− ( q + 11 ) − ( q + 11 )

− ( 3 x − 7 ) − ( 3 x − 7 )

− ( 5 p − 4 ) − ( 5 p − 4 )

5 + 9 ( n − 6 ) 5 + 9 ( n − 6 )

12 + 8 ( u − 1 ) 12 + 8 ( u − 1 )

16 − 3 ( y + 8 ) 16 − 3 ( y + 8 )

18 − 4 ( x + 2 ) 18 − 4 ( x + 2 )

4 − 11 ( 3 c − 2 ) 4 − 11 ( 3 c − 2 )

9 − 6 ( 7 n − 5 ) 9 − 6 ( 7 n − 5 )

22 − ( a + 3 ) 22 − ( a + 3 )

8 − ( r − 7 ) 8 − ( r − 7 )

−12 − ( u + 10 ) −12 − ( u + 10 )

−4 − ( c − 10 ) −4 − ( c − 10 )

( 5 m − 3 ) − ( m + 7 ) ( 5 m − 3 ) − ( m + 7 )

( 4 y − 1 ) − ( y − 2 ) ( 4 y − 1 ) − ( y − 2 )

5 ( 2 n + 9 ) + 12 ( n − 3 ) 5 ( 2 n + 9 ) + 12 ( n − 3 )

9 ( 5 u + 8 ) + 2 ( u − 6 ) 9 ( 5 u + 8 ) + 2 ( u − 6 )

9 ( 8 x − 3 ) − ( −2 ) 9 ( 8 x − 3 ) − ( −2 )

4 ( 6 x − 1 ) − ( −8 ) 4 ( 6 x − 1 ) − ( −8 )

14 ( c − 1 ) − 8 ( c − 6 ) 14 ( c − 1 ) − 8 ( c − 6 )

11 ( n − 7 ) − 5 ( n − 1 ) 11 ( n − 7 ) − 5 ( n − 1 )

6 ( 7 y + 8 ) − ( 30 y − 15 ) 6 ( 7 y + 8 ) − ( 30 y − 15 )

7 ( 3 n + 9 ) − ( 4 n − 13 ) 7 ( 3 n + 9 ) − ( 4 n − 13 )

In the following exercises, evaluate both expressions for the given value.

If v = −2 , v = −2 , evaluate

• ⓐ 6 ( 4 v + 7 ) 6 ( 4 v + 7 )
• ⓑ 6 · 4 v + 6 · 7 6 · 4 v + 6 · 7

If u = −1 , u = −1 , evaluate

• ⓐ 8 ( 5 u + 12 ) 8 ( 5 u + 12 )
• ⓑ 8 · 5 u + 8 · 12 8 · 5 u + 8 · 12

If n = 2 3 , n = 2 3 , evaluate

• ⓐ 3 ( n + 5 6 ) 3 ( n + 5 6 )
• ⓑ 3 · n + 3 · 5 6 3 · n + 3 · 5 6

If y = 3 4 , y = 3 4 , evaluate

• ⓐ 4 ( y + 3 8 ) 4 ( y + 3 8 )
• ⓑ 4 · y + 4 · 3 8 4 · y + 4 · 3 8

If y = 7 12 , y = 7 12 , evaluate

• ⓐ −3 ( 4 y + 15 ) −3 ( 4 y + 15 )
• ⓑ −3 · 4 y + ( −3 ) · 15 −3 · 4 y + ( −3 ) · 15

If p = 23 30 , p = 23 30 , evaluate

• ⓐ −6 ( 5 p + 11 ) −6 ( 5 p + 11 )
• ⓑ −6 · 5 p + ( −6 ) · 11 −6 · 5 p + ( −6 ) · 11

If m = 0.4 , m = 0.4 , evaluate

• ⓐ −10 ( 3 m − 0.9 ) −10 ( 3 m − 0.9 )
• ⓑ −10 · 3 m − ( −10 ) ( 0.9 ) −10 · 3 m − ( −10 ) ( 0.9 )

If n = 0.75 , n = 0.75 , evaluate

• ⓐ −100 ( 5 n + 1.5 ) −100 ( 5 n + 1.5 )
• ⓑ −100 · 5 n + ( −100 ) ( 1.5 ) −100 · 5 n + ( −100 ) ( 1.5 )

If y = −25 , y = −25 , evaluate

• ⓐ − ( y − 25 ) − ( y − 25 )
• ⓑ − y + 25 − y + 25

If w = −80 , w = −80 , evaluate

• ⓐ − ( w − 80 ) − ( w − 80 )
• ⓑ − w + 80 − w + 80

If p = 0.19 , p = 0.19 , evaluate

• ⓐ − ( p + 0.72 ) − ( p + 0.72 )
• ⓑ − p − 0.72 − p − 0.72

If q = 0.55 , q = 0.55 , evaluate

• ⓐ − ( q + 0.48 ) − ( q + 0.48 )
• ⓑ − q − 0.48 − q − 0.48

Everyday Math

Buying by the case Joe can buy his favorite ice tea at a convenience store for $1.99 $1.99 per bottle. At the grocery store, he can buy a case of 12 12 bottles for $23.88 . $23.88 .

ⓐ Use the distributive property to find the cost of 12 12 bottles bought individually at the convenience store. (Hint: notice that $1.99 $1.99 is $2 − $0.01 . $2 − $0.01 . )

ⓑ Is it a bargain to buy the iced tea at the grocery store by the case?

Multi-pack purchase Adele’s shampoo sells for $3.97 $3.97 per bottle at the drug store. At the warehouse store, the same shampoo is sold as a 3-pack 3-pack for $10.49 . $10.49 .

ⓐ Show how you can use the distributive property to find the cost of 3 3 bottles bought individually at the drug store.

ⓑ How much would Adele save by buying the 3-pack 3-pack at the warehouse store?

Writing Exercises

Simplify 8 ( x − 1 4 ) 8 ( x − 1 4 ) using the distributive property and explain each step.

Explain how you can multiply 4 ( $5.97 ) 4 ( $5.97 ) without paper or a calculator by thinking of $5.97 $5.97 as 6 − 0.03 6 − 0.03 and then using the distributive property.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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7.4: Distributive Property

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Learning Objectives

• Simplify expressions using the distributive property
• Evaluate expressions using the distributive property

be prepared!

Before you get started, take this readiness quiz.

• Multiply: 3(0.25). If you missed this problem, review Example 5.3.5
• Simplify: 10 − (−2)(3). If you missed this problem, review Example 3.7.5 .
• Combine like terms: 9y + 17 + 3y − 2. If you missed this problem, review Example 2.3.10 .

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $9.25; that is, 9 dollars and 1 quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

They need 3 times $9, so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the Distributive Property.

Definition: Distributive Property

If a, b, c are real numbers, then a(b + c) = ab + ac.

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

\[\begin{split} 3(9&.25) \\ 3(9 &+ 0.25) \\ 3(9) &+ 3(0.25) \\ 27 &+ 0.75 \\ 27&.75 \end{split}\]

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression 3(x + 4), the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the Distributive Property, as shown in Example \(\PageIndex{1}\).

Example \(\PageIndex{1}\):

Simplify: 3(x + 4).

Exercise \(\PageIndex{1}\):

Simplify: 4(x + 2).

Exercise \(\PageIndex{2}\):

Simplify: 6(x + 7).

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 7.17 would look like this:

\[3 \cdot x + 3 \cdot 4\]

Example \(\PageIndex{2}\):

Simplify: 6(5y + 1).

Exercise \(\PageIndex{3}\):

Simplify: 9(3y + 8).

Exercise \(\PageIndex{4}\):

Simplify: 5(5w + 9).

The distributive property can be used to simplify expressions that look slightly different from a(b + c). Here are two other forms.

If a, b, c are real numbers, then\[a(b + c) = ab + ac$$Other forms$$a(b − c) = ab − ac$$$$(b + c)a = ba + ca\]

Example \(\PageIndex{3}\):

Simplify: 2(x − 3).

Exercise \(\PageIndex{5}\):

Simplify: 7(x − 6).

Exercise \(\PageIndex{6}\):

Simplify: 8(x − 5).

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Example \(\PageIndex{4}\):

Simplify: \(\dfrac{3}{4}\)(n + 12).

Exercise \(\PageIndex{7}\):

Simplify: \(\dfrac{2}{5}\)(p + 10).

\(\frac{2}{5}p + 4 \)

Exercise \(\PageIndex{8}\):

Simplify: \(\dfrac{3}{7}\)(u + 21).

\(\frac{3}{7}u +9 \)

Example \(\PageIndex{5}\):

Simplify: \(8 \left(\dfrac{3}{8}x + \dfrac{1}{4}\right)\).

Exercise \(\PageIndex{9}\):

Simplify: \(6 \left(\dfrac{5}{6}y + \dfrac{1}{2}\right)\).

Exercise \(\PageIndex{10}\):

Simplify: \(12 \left(\dfrac{1}{3}n + \dfrac{3}{4}\right)\).

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

Example \(\PageIndex{6}\):

Simplify: 100(0.3 + 0.25q).

Exercise \(\PageIndex{11}\):

Simplify: 100(0.7 + 0.15p).

Exercise \(\PageIndex{12}\):

Simplify: 100(0.04 + 0.35d).

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

Example \(\PageIndex{7}\):

Simplify: \(m(n − 4)\).

Notice that we wrote m • 4 as 4m. We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

Exercise \(\PageIndex{13}\):

Simplify: r(s − 2).

Exercise \(\PageIndex{14}\):

Simplify: y(z − 8).

The next example will use the ‘backwards’ form of the Distributive Property, (b + c)a = ba + ca.

Example \(\PageIndex{8}\):

Simplify: (x + 8)p.

Exercise \(\PageIndex{15}\):

Simplify: (x + 2)p.

Exercise \(\PageIndex{16}\):

Simplify: (y + 4)q.

When you distribute a negative number, you need to be extra careful to get the signs correct.

Example \(\PageIndex{9}\):

Simplify: −2(4y + 1).

Exercise \(\PageIndex{17}\):

Simplify: −3(6m + 5).

Exercise \(\PageIndex{18}\):

Simplify: −6(8n + 11).

Example \(\PageIndex{10}\):

Simplify: −11(4 − 3a).

You could also write the result as 33a − 44. Do you know why?

Exercise \(\PageIndex{19}\):

Simplify: −5(2 − 3a).

Exercise \(\PageIndex{20}\):

Simplify: −7(8 − 15y).

In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, −a = −1 • a.

Example \(\PageIndex{11}\):

Simplify: −(y + 5).

Exercise \(\PageIndex{21}\):

Simplify: −(z − 11).

Exercise \(\PageIndex{22}\):

Simplify: −(x − 4).

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Example \(\PageIndex{12}\):

Simplify: 8 − 2(x + 3).

Exercise \(\PageIndex{23}\):

Simplify: 9 − 3(x + 2).

Exercise \(\PageIndex{24}\):

Simplify: 7x − 5(x + 4).

Example \(\PageIndex{13}\):

Simplify: 4(x − 8) − (x + 3).

Exercise \(\PageIndex{25}\):

Simplify: 6(x − 9) − (x + 12).

Exercise \(\PageIndex{26}\):

Simplify: 8(x − 1) − (x + 5).

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works. In the examples below, we will practice evaluating some of the expressions from previous examples; in part (a), we will evaluate the form with parentheses, and in part (b) we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example \(\PageIndex{14}\):

When y = 10 evaluate: (a) 6(5y + 1) (b) 6 • 5y + 6 • 1.

(a) 6(5y + 1)

(b) 6 • 5y + 6 • 1

Notice, the answers are the same. When y = 10, 6(5y + 1) = 6 • 5y + 6 • 1. Try it yourself for a different value of y.

Exercise \(\PageIndex{27}\):

Evaluate when w = 3: (a) 5(5w + 9) (b) 5 • 5w + 5 • 9.

Exercise \(\PageIndex{28}\):

Evaluate when y = 2: (a) 9(3y + 8) (b) 9 • 3y + 9 • 8.

Example \(\PageIndex{15}\):

When y = 3, evaluate (a) −2(4y + 1) (b) −2 • 4y + (−2) • 1.

(a) −2(4y + 1)

(b) −2 • 4y + (−2) • 1

Exercise \(\PageIndex{29}\):

Evaluate when n = −2: (a) −6(8n + 11) (b) −6 • 8n + (−6) • 11.

Exercise \(\PageIndex{30}\):

Evaluate when m = −1: (a) −3(6m + 5) (b) −3 • 6m + (−3) • 5.

Example \(\PageIndex{16}\):

When y = 35 evaluate (a) −(y + 5) and (b) −y − 5 to show that −(y + 5) = −y − 5.

(a) −(y + 5)

(b) −y − 5

Exercise \(\PageIndex{31}\):

Evaluate when x = 36: (a) −(x − 4) (b) −x + 4 to show that −(x − 4) = − x + 4.

Exercise \(\PageIndex{32}\):

Evaluate when z = 55: (a) −(z − 10) (b) −z + 10 to show that −(z − 10) = − z + 10.

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The Distributive Property

Practice Makes Perfect

In the following exercises, simplify using the distributive property.

• 6(c − 13)
• 7(y − 13)
• 7(3p − 8)
• 5(7u − 4)
• \(\dfrac{1}{2}\)(n + 8)
• \(\dfrac{1}{3}\)(u + 9)
• \(\dfrac{1}{4}\)(3q + 12)
• \(\dfrac{1}{5}\)(4m + 20)
• \(9 \left(\dfrac{5}{9} y − \dfrac{1}{3}\right)\)
• \(10 \left(\dfrac{3}{10} x − \dfrac{2}{5}\right)\)
• \(12 \left(\dfrac{1}{4} + \dfrac{2}{3} r\right)\)
• \(12 \left(\dfrac{1}{6} + \dfrac{3}{4} s\right)\)
• r(s − 18)
• u(v − 10)
• −2(y + 13)
• −3(a + 11)
• −7(4p + 1)
• −9(9a + 4)
• −3(x − 6)
• −4(q − 7)
• −9(3a − 7)
• −6(7x − 8)
• −(r + 7)
• −(q + 11)
• −(3x − 7)
• −(5p − 4)
• 5 + 9(n − 6)
• 12 + 8(u − 1)
• 16 − 3(y + 8)
• 18 − 4(x + 2)
• 4 − 11(3c − 2)
• 9 − 6(7n − 5)
• 22 − (a + 3)
• 8 − (r − 7)
• −12 − (u + 10)
• −4 − (c − 10)
• (5m − 3) − (m + 7)
• (4y − 1) − (y − 2)
• 5(2n + 9) + 12(n − 3)
• 9(5u + 8) + 2(u − 6)
• 9(8x − 3) − (−2)
• 4(6x − 1) − (−8)
• 14(c − 1) − 8(c − 6)
• 11(n − 7) − 5(n − 1)
• 6(7y + 8) − (30y − 15)
• 7(3n + 9) − (4n − 13)

In the following exercises, evaluate both expressions for the given value.

• 6 · 4v + 6 · 7
• 8 · 5u + 8 · 12
• \(3 \left(n + \dfrac{5}{6}\right)\)
• 3 • n + 3 • \(\dfrac{5}{6}\)
• 4 ⎛ ⎝ y + 3 8 ⎞ ⎠
• 4 • y + 4 • \(\dfrac{3}{8}\)
• −3(4y + 15)
• 3 • 4y + (−3) • 15
• −6(5p + 11)
• −6 • 5p + (−6) • 11
• −10(3m − 0.9)
• −10 • 3m − (−10)(0.9)
• −100(5n + 1.5)
• −100 • 5n + (−100)(1.5)
• −(y − 25)
• −y + 25
• −(w − 80)
• −w + 80
• −(p + 0.72)
• −p − 0.72
• −(q + 0.48)
• −q − 0.48

Everyday Math

• Use the distributive property to find the cost of 12 bottles bought individually at the convenience store. (Hint: notice that $1.99 is $2 − $0.01.)
• Is it a bargain to buy the iced tea at the grocery store by the case?
• Show how you can use the distributive property to find the cost of 3 bottles bought individually at the drug store.
• How much would Adele save by buying the 3-pack at the warehouse store?

Writing Exercises

• Simplify \(8 \left(x − \dfrac{1}{4}\right)\) using the distributive property and explain each step.
• Explain how you can multiply 4($5.97) without paper or a calculator by thinking of $5.97 as 6 − 0.03 and then using the distributive property.

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

Distributive Property in Algebra

Related Topics: More Lessons for Grade 7 Math Worksheets

Videos, worksheets, stories and songs to help Grade 7 students learn how to use the Distributive Property in algebra.

The distributive property of addition and multiplication states that multiplying a sum by a number is the same as multiplying each addend by that number and then adding the two products.

3( x + 4) = 3 × x + 3 × 4 = 3 x + 12

The Distributive Property. One of the most important and foundational topics in basic mathematics. The concept is explained and example problems are worked out and explained in detail.

Using the distributive property in algebra.

0:00 Definition of the Distributive Property 12:15 Modeling the Distributive Property 18:45 Practice the Distributive Property 28:00 Factor using the Distributive Property

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Distributive property states that a term in an algebraic expression is multiplied to all the terms present inside the bracket. Distributive property practice problems are the best for students to practice breaking down difficult equations into parts to easily solve them and find the answer. It also helps students to understand applications of distributive property.

Teachers can use these problems in a classroom to enhance students' understanding.

You can also try our Distributive Property Worksheet and Distributive property (expanding) as well for a better understanding of the concepts.

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Distributive Property Practice Problems

By: Author Sarah Carter

Posted on Published: September 29, 2017  - Last updated: July 23, 2022

Categories INBs , Distributive Property

After completing our distributive property foldable , I wanted to give my students a bit more practice. That’s how this set of distributive property practice problems was born.

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More Activities and Resources for Teaching the Distributive Property

Friday 29th of September 2017

I love!!! that grouping strategy!! I would have my students box one set of like terms and put circles around another and triangles around another and often times they would write the shape through the sign of the number and then end up messing up. I love how you had them just list them. Also by using a few big problems they really see everything they need to group together. Great idea, and thank you!!

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Distributive property

Here you will learn about the distributive property, including what it is, and how to use it to solve problems.

Students will first learn about the distributive property as part of operations and algebraic thinking in 3rd grade.

What is the distributive property?

The distributive property states that multiplying the sum of two or more numbers is the same as multiplying the addends separately.

For example,

When multiplying 2 \times 8, you can break 8 up into 2 + 6.

The distributive property says that you can multiply the parts separately and then add the products together.

Any way you solve the equivalent expressions, the product is the same.

For most expressions, there is more than one way to use the distributive property. 

When multiplying 2 \times 8, you can break 8 up into 5 + 3.

Common Core State Standards

How does this relate to 3rd grade math?

• Grade 3 – Operations and Algebraic Thinking (3.OA.B.5) Apply properties of operations as strategies to multiply and divide. Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication.) 3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication.) Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Distributive property.)

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How to use the distributive property

In order to use the distributive property:

Identify an equation multiplying two numbers.

Show one of the numbers being multiplied as a sum of numbers.

Multiply each number in the sum.

Add the partial products together to find the final product.

Distributive property examples

Example 1: distributive property with basic facts.

Show how to solve 3 \times 5 using the distributive property.

You can use the distributive property with 3 \times 5, since it is multiplication.

2 Show one of the numbers being multiplied as a sum of numbers.

Either number can be used, but for this example let’s break up 5 into 4 + 1.

3 \times 5=3 \times(4+1)

3 Multiply each number in the sum.

\begin{aligned} & 3 \times(4+1) \\\\ & =(3 \times 4)+(3 \times 1) \\\\ & =12+3 \end{aligned}

4 Add the partial products together to find the final product.

12 + 3 = 15

3 \times 5=15 can be solved using the distributive property.

Example 2: distributive property with basic facts

Show how to solve 12 \times 9 using the distributive property.

You can use the distributive property with 12 \times 9, since it is multiplication.

Either number can be used, but for this example let’s break up 9 into 3 + 3 + 3.

12 \times 9=12 \times(3+3+3)

\begin{aligned} & 12 \times(3+3+3) \\\\ & =(12 \times 3)+(12 \times 3)+(12 \times 3) \\\\ & =36+36+36 \end{aligned}

36 + 36 + 36 = 108

12 \times 9=108 can be solved using the distributive property.

Example 3: distributive property with basic facts

Show how to solve 7 \times 6 using the distributive property.

You can use the distributive property with 7 \times 6, since it is multiplication.

Either number can be used, but for this example let’s break up 7 into 4 + 3.

7 \times 6=(4+3) \times 6

\begin{aligned} & (4+3) \times 6 \\\\ & =(4 \times 6)+(3 \times 6) \\\\ & =24+18 \end{aligned}

24 + 18 = 42

7 \times 6=42 can be solved using the distributive property.

Example 4: distributive property with basic facts

Show how to solve 4 \times 11 using the distributive property.

You can use the distributive property with 4 \times 11, since it is multiplication.

Either number can be used, but for this example let’s break up 11 into 10 + 1.

4 \times 11=4 \times(10+1)

\begin{aligned} & 4 \times(10+1) \\\\ & =(4 \times 10)+(4 \times 1) \\\\ & =40+4 \end{aligned}

40 + 4 = 44

4 \times 11=44 can be solved using the distributive property.

Example 5: distributive property with basic facts

Show how to solve 8 \times 5 using the distributive property.

You can use the distributive property with 8 \times 5, since it is multiplication.

Either number can be used, but for this example let’s break up 8 into 2 + 6.

8 \times 5=(2+6) \times 5

\begin{aligned} & (2+6) \times 5 \\\\ & =(2 \times 5)+(6 \times 5) \\\\ & =10+30 \end{aligned}

10 + 30 = 40

8 \times 5=40 can be solved using the distributive property.

Example 6: distributive property with basic facts

Show how to solve 3 \times 12 using the distributive property.

You can use the distributive property with 3 \times 12, since it is multiplication.

Either number can be used, but for this example let’s break up 12 into 1 + 1 + 10.

3 \times 12=3 \times(1+1+10)

\begin{aligned} & 3 \times(1+1+10) \\\\ & =(3 \times 1)+(3 \times 1)+(3 \times 10) \\\\ & =3+3+30 \end{aligned}

3 + 3 + 30 = 36

3 \times 12=36 can be solved using the distributive property.

Teaching tips for the distributive property

• Intentionally choose practice problems that lend themselves to being solved with the distributive property, as it is not always necessary or useful in all solving situations.
• Instead of just giving students the distributive property definition, draw attention to examples of the distributive property as they come up in daily math activities. You may even keep an anchor chart of different examples. Over time, students will start using it and recognizing it on their own and then you can introduce them to the property and its official definition through their own examples.
• Include plenty of student discourse around this topic to ensure that students understand that breaking apart a number and then multiplying it in parts does not change the total product. This could include students sharing their thinking or critiquing the thinking of others.

Easy mistakes to make

• Thinking there is only one way to use the distributive property to solve Often, there is more than one way to use the distributive property when solving. For example, \begin{aligned} & 4 \times 5 \hspace{4.65cm} 4 \times 5 \\ & =(2+2) \times 5 \hspace{3.5cm} =(1+3) \times 5 \\ & =(2 \times 5)+(2 \times 5) \hspace{1cm} \text{ OR } \hspace{1cm} =(1 \times 5)+(3 \times 5) \\ & =10+10 \hspace{3.9cm} =5+15 \\ & =20 \hspace{4.63cm} =20 \end{aligned}

Related properties of equality lessons

• Properties of equality
• Order of operations
• Associative property
• Commutative property

Practice distributive property questions

1. Which of the following equations shows 12 \times 6 using the distributive property?

The numbers are being multiplied, so the distributive property can be used.

2. Which of the following equations shows 7 \times 9 using the distributive property?

3. Which of the following equations shows 11 \times 8 using the distributive property?

4. Which of the following equations shows 3 \times 7 using the distributive property?

5. Which of the following equations is NOT a way to solve 10 \times 5 using the distributive property?

This strategy is NOT a way to solve with the distributive property.

All the other equations break 10 or 5 up into a sum and add the products of the parts, using the distributive property correctly:

6. Which of the following equations is NOT a way to solve 9 \times 8 using the distributive property?

This strategy is NOT a way to solve 9 \times 8 with the distributive property.

All the other equations break 9 or 8 up into a sum and add the products of the parts, using the distributive property correctly:

Distributive property FAQs

Yes, the distributive property can be used with integers (including negative numbers) and rational numbers (including fractions and decimals), as long as the numbers are all being multiplied. In middle and high school, students will learn how to use the distributive property with any real number and/or algebraic expression.

No, even though the associative property also uses parentheses, they are different properties. The associative property says you can change the grouping of numbers when adding or multiplying and the sum or product will be the same. This is different from the distributive property.

Yes, this is called the distributive property of multiplication over subtraction.

This is a general term and means the same as the distributive property.

No, because of the order of operations (or PEMDAS), the products will be found first and then added together. However, it is good practice to group each partial product with parentheses.

The next lessons are

• Addition and subtraction
• Multiplication and division
• Types of numbers

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Distributive Property – Definition, Examples, Facts, FAQs

What is distributive property, verification of the distributive property, how to use the distributive property, solved examples on distributive property, practice problems on distributive property, frequently asked questions on distributive property.

According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

To “distribute” means to divide something or give a share or part of something.

So what does distributive property mean in math? The distributive property describes how we can distribute multiplication over addition and subtraction.

Distributive Property Formula (Bold)

According to the distributive property, an expression of the form $A (B + C)$ can be solved as

$A (B + C) = AB + AC$.

This property applies to subtraction as well. 

$A (B \;–\; C) = AB \;–\; AC$

Here, the multiplier A is ‘distributed’ in order to multiply it with B and C separately. 

(A, B, and C are any real numbers.)

Let’s visualize the distributive property using multiplication arrays!

$3 \times (2 + 3)$ is represented by 3 rows and 5 columns. 

$3 \times 2$ is represented by 3 rows and 2 columns.

$3 \times 3$ is represented by 3 rows and 3 columns.

You can see that we broke down a big array into two smaller arrays using the distributive property.

Here’s an example of how the result does not change when solved normally and when solved using the distributive property:

This property helps in making difficult problems simpler. You can use this property of multiplication to rewrite an expression by distributing or breaking down a factor as a sum or difference of two numbers.

Distributive Property: Definition

The distributive property is a fundamental property that defines how multiplication operation is distributed over addition and subtraction .

The distributive property is also called the distributive law of multiplication over addition and subtraction. 

Related Worksheets

Distributive Property of Multiplication over Addition

When we have to multiply a number by the sum of two numbers, we use the distributive property of multiplication over addition. 

Distributive Property of Multiplication over Addition:  $A (B + C) = AB + AC$

Distributive Property of Multiplication over Subtraction

The distributive property of multiplication over subtraction is equivalent to the distributive property of multiplication over addition, except for the operations of addition and subtraction.

Distributive Property of Multiplication over Subtraction:  $A (B \;-\; C) = AB \;-\; AC$

$A(B \;−\; C)$ and $AB \;−\; AC$ are equivalent expressions.

Consider these distributive property examples below.

We can verify the distributive property by solving both LHS and RHS. 

Example : Solve the expression $2 (4 \;–\; 3)$. 

Using the distributive law of multiplication over subtraction, we have

$2 \times (4 \;–\; 3) = (2 \times 4) \;–\; 2 \times 3 = 8 \;–\; 6 = 2$

Again, if we try to solve the expression with the order of operations or PEMDAS, we’ll have to subtract the numbers in parentheses, then multiply the difference with the number outside the parentheses, which implies:

$2 \times (4 \;–\; 3)=2 \times 1 = 2$

The distributive property of subtraction is verified since both techniques give the same result.

There are three simple steps to use the distributive property.

Step 1: Distribute the multiplier (the number outside the parentheses).

Step 2: Find the individual products.

Step 3: Add or subtract.

Let’s understand how to use the distributive property with the help of examples.

Examples of distributive property of multiplication over addition and subtraction:

Example 1: Solve the expression: $6 \times (20 + 5)$ using the distributive property of multiplication over addition. 

Let’s use the property to calculate the expression $6 \times (20 + 5)$, the number 6 is spread across the two addends. To put it simply, we multiply each addend by 6 and then the products can be added.

$6 \times 20 + 6 \times 5 = 120 + 30 = 150$

Example 2: Solve the expression $2 \times (2 + 4)$ using the distributive law of multiplication over addition.

$2 \times (2 + 4) = 2 \times 2 + 2 \times 4 = 4 + 8 = 12$

If we try to solve this expression using the PEMDAS rule, we’ll have to add the numbers in parentheses and then multiply the total by the number outside the parentheses. This implies:

$2 (2 + 4) = 2 \times 6 = 12$

Thus, we get the same result irrespective of the method used.

Example 3: Solve the expression $6 \times (20 \;–\; 5)$ using the distributive property of multiplication over subtraction.

Using the distributive property of multiplication over subtraction, we get

$6 \times (20 \;–\; 5) = 6 \times 20 \;–\; 6 \times 5 = 120 \;–\; 30 = 90$

Distributive Property of Division

We can use the distributive law of division by distributing the dividend (or breaking down the dividend) into the sum of two numbers (partial dividends), such that both the numbers are completely divisible by the divisor. 

Example: $132 \div 6$

132 can be broken down as $60 + 60 + 12$, thus making division easier. Note that 60 and 12 both are divisible by 6.

$132 \div 6 = (60 + 60 + 12) \div 6$

$132 \div 6 = (60 \div 6) + (60 \div 6) + (12 \div 6)$

$132 \div 6 = 10 + 10 + 2$

$132 \div 6 = 22$

We cannot break $132 \div 6$ as $(50 + 50 + 32)\div 6$ since 50 and 32 are not divisible by 6.

Also, we cannot break the divisor:  $132 \div (4 + 2)$ will give you the wrong result. 

Facts about Distributive Property

• We can describe the distributive property as breaking down a multiplication fact into the sum of two multiplication facts.
• You can also use the distributive property with variables when simplifying, expanding, polynomial expressions.

In this article, we learned about the distributive property, formulas, when to use the distributive property, and also how to use distributive property in complex equations and problems. Let’s solve a few examples and practice problems based on the distributive property.

Example 1:  Solve $(5 + 7 + 3) 4$ .

Solution : Using the distributive property of multiplication over addition,

$A \times (B + C) = AB + AC$

$(5 + 7 + 3) \times 4$

$= ( 5 \times 4) + (7 \times 4) + (3 \times 4)$

$= 20 + 28 + 12$

We can verify it as:

$(5 + 7 + 3) \times 4 = 15 \times 4 = 60$

Example 2: Solve the following using the distributive property: $\;−\;2 (\;−\;x \;−\; 7)$ . 

Solution : Using the distributive property,

$\;−\;2 (\;−\;x \;−\; 7) = (\;−\;2)(\;−\;x) \;−\; (\;−\;2)(7)$

     $= 2x \;−\; (\;−\;14)$

     $= 2x + 14$

Example 3: Which property does the equation $3 (4 \;−\; 9) = (3 \times 4) \;−\; (3 \times 9)$ show?

Given equation: $3 (4 \;−\; 9) = (3 \times 4) \;−\; (3 \times 9)$

Comparing it with the distributive property formula for multiplication over subtraction $A (B \;-\; C) = AB \;-\; AC$, we get that the above equation shows the distributive property of multiplication over subtraction.

Distributive Property - Definition with Examples

Attend this quiz & Test your knowledge.

The expression $7 ($x$ + 6)$ equals

The expression $3 (7x \;–\; 8)$ equals, the expression m $(3$n $– 9)$ equals, the yield of a banana farm is 355 dozens of bananas. how many bananas were harvested.

Does distributive property apply to division too?

The distributive property applies to division in the same way that it applies to multiplication. However, the concept of “breaking apart” or “distributing” can be applied with division only by dividing the numerator into smaller amounts that are exactly divisible by the divisor. 

For example, to solve $\frac{125}{5}$, we can divide the numerator(125) as: (50 + 50 + 25)

therefore:$\frac{125}{5}$ = $\frac{50}{5}$ + $\frac{50}{5}$ + $\frac{25}{5}$ = 10 + 10 + 5 = 25.

What is the rule for the distributive property?

According to the distributive property, multiplying the sum of two or more addends by a number produces the same result as when each addend is multiplied individually by the number and the products are added together.

How can distributive property help in solving complex questions?

The distributive property distributes complex expressions into simpler terms and thus makes problems, especially with multiple factors, easier to solve.

Can parentheses be removed after distributing?

Yes, when applying the distributive property, the outside factor is multiplied by each term inside the parentheses. This gets rid of parentheses.

How do we use the distributive property with equations?

We can use the distributive property either to expand terms in a given expression or equation or to simplify it based on the requirement.

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• Operations on Rational Numbers – Methods, Steps, Facts, Examples
• Subtract Fractions with Unlike Denominators – Definition with Examples

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SAT Math : Distributive Property

Study concepts, example questions & explanations for sat math, all sat math resources, example questions, example question #1 : distributive property.

Example Question #1 : How To Use Foil In The Distributive Property

Remember that ( a –  b )( a +  b ) = a 2 –  b 2 .

We can therefore rewrite (3 x –  4)(3 x + 4) = 2 as (3 x ) 2  – (4) 2 = 2.

Simplify to find 9 x 2  – 16 = 2.

Adding 16 to each side gives us 9 x 2 = 18.

We are asked to find the difference between g(h(x)) and h(g(x)), where g(x) = 2x 2  – 2 and h(x) = x + 4. Let's find expressions for both.

g(h(x)) = g(x + 4) = 2(x + 4) 2  – 2

g(h(x)) = 2(x + 4)(x + 4) – 2

In order to find (x+4)(x+4) we can use the FOIL method.

(x + 4)(x + 4) = x 2 + 4x + 4x + 16

g(h(x)) = 2(x 2 + 4x + 4x + 16) – 2

g(h(x)) = 2(x 2 + 8x + 16) – 2

Distribute and simplify.

g(h(x)) = 2x 2 + 16x + 32 – 2

g(h(x)) = 2x 2 + 16x + 30

Now, we need to find h(g(x)).

h(g(x)) = h(2x 2  – 2) = 2x 2  – 2 + 4

h(g(x)) = 2x 2 + 2

Finally, we can find g(h(x)) – h(g(x)).

g(h(x)) – h(g(x)) = 2x 2 + 16x + 30 – (2x 2 + 2)

= 2x 2 + 16x + 30 – 2x 2  – 2

The answer is 16x + 28.

Example Question #4 : Distributive Property

Let the two numbers be  x and  y .

x +  y =  s

( x + 1)( y + 1) =  q

Expand the last equation:

xy +  x +  y + 1 =  q

Note that both of the first two equations can be substituted into this new equation:

p +  s + 1 =  q

Solve this equation for  q – p by subtracting p from both sides:

s + 1 =  q  –  p

Example Question #5 : Distributive Property

Expand the expression:

When using FOIL, multiply the first, outside, inside, then last expressions; then combine like terms.

Expand the following expression:

Which becomes

Or, written better

Example Question #2671 : Sat Mathematics

Multiply using FOIL:

First = 3x(2x) = 6x 2

Outter = 3x(4) = 12x

Inner = -1(2x) = -2x

Last = -1(4) = -4

Combine and simplify:

6x 2 + 12x - 2x - 4 = 6x 2 +10x - 4

Simplify the expression.

None of the other answers

Solve by applying FOIL:

First: 2x 2 * 2y = 4x 2 y

Outer: 2x 2 * a = 2ax 2

Inner: –3x * 2y = –6xy

Last: –3x * a = –3ax

Add them together: 4x 2 y + 2ax 2 – 6xy – 3ax

There are no common terms, so we are done.

Example Question #2 : How To Use Foil In The Distributive Property

Use FOIL to expand the left side of the equation.

Expand and simplify the expression.

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Associative, Distributive and Commutative Properties

All together on 1 page!

To learn more about any of the properties below, visit that property's individual page.

Properties and Operations

Let's look at how (and if) these properties work with addition , multiplication , subtraction and division.

Multiplication

Subtraction, practice problems.

Which of the following statements illustrate the distributive, associate and the commutative property?

Directions: Click on each answer button to see what property goes with the statement on the left .

All three of these properties can also be applied to Algebraic Expressions.

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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