Explainer: Rewriting Expressions Using the Distributive Property

In this explainer, we will learn how to rewrite algebraic expressions using the distributive property.

Distributive Property of Multiplication over Addition

The distributive property of multiplication over addition allows us to transform a product involving a sum in a sum of products.

The following equations demonstrate the distributive property of multiplication over addition.

In everyday language, to distribute means to spread or scatter over an area. Here, the idea is that the multiplication by π‘Ž will indeed occur over all the terms inside the brackets. So, the factor π‘Ž will be β€œdistributed” (multiplied) to each term inside the brackets, be the brackets located after or before π‘Ž.

The distributive property can be visualized with a diagram showing a rectangle of sides π‘Ž and (𝑏+𝑐).

The area of the rectangle is the product of its length and its width, that is, π‘Žβ‹…(𝑏+𝑐). By realizing that the larger rectangle is made of the two smaller rectangles, we see that the area of the larger rectangle is equal to π‘Žπ‘+π‘Žπ‘.

Hence, π‘Ž(𝑏+𝑐)=π‘Žπ‘+π‘Žπ‘.

It is obvious that the area of the rectangle does not depend on the order in which we multiply the length and the width, so we have π‘Ž(𝑏+𝑐)=(𝑏+𝑐)π‘Ž.

This property is called the commutative property of multiplication.

Let us have a look at what happens when there is a subtraction inside the brackets, that is, 𝑙(π‘šβˆ’π‘›).

The area of the shaded rectangle is 𝑙(π‘šβˆ’π‘›). We can say that it is equal to the area of the larger rectangle of sides 𝑙 and π‘š minus the area 𝑙𝑛. We see that 𝑙(π‘šβˆ’π‘›)=π‘™π‘šβˆ’π‘™π‘›.

The product involving a difference is indeed rewritten as the difference of the products.

Distributive Property of Multiplication over Subtraction

The distributive property of multiplication over subtraction allows us to transform the product involving a difference in a difference of products.

The following equations demonstrate the distributive property of multiplication over subtraction.

Example 1: Using the Distributive Property of Multiplication over Addition

Use the properties of real numbers to rewrite 6(π‘₯+4) as an equivalent expression that does not contain parentheses.

Answer

The algebraic expression 6(π‘₯+4) is a multiplication of two factors: 6 and (π‘₯+4). The distributive property allows us to transform this multiplication involving a sum into a sum of products by considering that the 6 is β€œdistributed” over all the terms in the brackets.

Therefore, we have 6(π‘₯+4)=6β‹…π‘₯+6β‹…4=6π‘₯+24.

Hence, our answer is 6(π‘₯+4)=6π‘₯+24.

Example 2: Using the Distributive Property of Multiplication over Subtraction

Use the distributive property to rewrite the algebraic expression 4(π‘₯βˆ’9).

Answer

The algebraic expression 4(π‘₯βˆ’9) is a multiplication of two factors: 4 and (π‘₯βˆ’9). The distributive property allows us to transform this multiplication involving a difference into a subtraction by considering that the 4 is β€œdistributed” over all the terms in the brackets.

It means that 4 is multiplied by π‘₯ (giving 4π‘₯) and by 9 (giving 36), and because there is a minus sign inside the brackets, meaning that 9 is taken away from π‘₯, then 36 is taken away from 4π‘₯.

Hence, we find that 4(π‘₯βˆ’9)=4π‘₯βˆ’36.

Example 3: Using the Distributive Property with Negative Numbers

Using the distributive property, rewrite the expression (βˆ’2π‘₯βˆ’8)(βˆ’6).

Answer

The expression (βˆ’2π‘₯βˆ’8)(βˆ’6) is a multiplication of two factors: (βˆ’2π‘₯βˆ’8) and βˆ’6. The distributive property allows us to rewrite this expression by considering that each term inside the brackets is multiplied by βˆ’6. Since the operation inside the brackets is a subtraction, the two terms inside the brackets will still be subtracted one from the other after they have been multiplied by βˆ’6.

Hence, we have (βˆ’2π‘₯βˆ’8)(βˆ’6)=12π‘₯+48.

Example 4: Identifying the Use of the Distributive Property

Which of the following demonstrates the distributive property?

  1. βˆ’4+(π‘₯+3)=3+(π‘₯βˆ’4)
  2. βˆ’4π‘₯+3=3βˆ’4π‘₯
  3. βˆ’4π‘₯+3+0=βˆ’4π‘₯+3
  4. βˆ’4(π‘₯+3)=βˆ’4π‘₯βˆ’12

Answer

The distributive property allows us to transform an algebraic expression involving a product where at least one of the factors consists of a sum or a difference of terms in a sum or a difference of products. In the question, only Option D contains a product where one of the factors is a sum of terms, namely, βˆ’4(π‘₯+3), which is the product of βˆ’4 and (π‘₯+3). The distributive property means that the factor βˆ’4 is distributed over the two terms inside the brackets; that is, βˆ’4(π‘₯+3)=(βˆ’4)β‹…π‘₯+(βˆ’4)β‹…3.

Carrying out the products in the right-hand side of the equation and writing βˆ’12 for +(βˆ’12), we find indeed that βˆ’4(π‘₯+3)=βˆ’4π‘₯βˆ’12.

Hence, the answer is option D.

Example 5: Using the Distributive Property in a Word Problem

Michael had 𝑏 dollars in his savings account and then he deposited $11. Seven months later, his balance had doubled. Which of the following is equivalent to his new balance of 2(𝑏+11)dollars?

  1. 2𝑏+11
  2. 𝑏+22
  3. 2𝑏+13
  4. 2𝑏+22
  5. 𝑏+13

Answer

Michael had 𝑏 dollars to which he added $11, which means that he then had 𝑏+11.dollars

We are told that this amount doubled after seven months; that is, he then had 2(𝑏+11).dollars

We can, here, apply the distributive property of the multiplication over addition, which means that we multiply the factor 2 by each term inside the brackets and the products together:

Hence, we find that 2(𝑏+11)=2𝑏+22.

The answer is Option D.

Key Points

  • The distributive property enables us to rewrite algebraic expressions without using parentheses.
  • Multiplication can be distributed over both addition and subtraction.
  • The distributive property of multiplication over addition allows us to transform a product involving a sum into a sum of products.
    The following equations demonstrate the distributive property of multiplication over addition.
  • The distributive property of multiplication over subtraction allows us to transform the product involving a difference into a difference of products.
    The following equations demonstrate the distributive property of multiplication over subtraction.

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