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 will indeed occur over all the terms inside the brackets. So, the factor will be “distributed” (multiplied) to each term inside the brackets, be the brackets located after or before .
The distributive property can be visualized with a diagram showing a rectangle of sides and .
The area of the rectangle is the product of its length and its width, that is, . 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
Hence,
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
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, .
The area of the shaded rectangle is . We can say that it is equal to the area of the larger rectangle of sides and minus the area . We see that
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 as an equivalent expression that does not contain parentheses.
Answer
The algebraic expression is a multiplication of two factors: 6 and . The distributive property allows us to transform this multiplication involving a sum into a sum of products by considering that the 6 is “distributed” over all the terms in the brackets.
Therefore, we have
Hence, our answer is
Example 2: Using the Distributive Property of Multiplication over Subtraction
Use the distributive property to rewrite the algebraic expression .
Answer
The algebraic expression is a multiplication of two factors: 4 and . The distributive property allows us to transform this multiplication involving a difference into a subtraction by considering that the 4 is “distributed” over all the terms in the brackets.
It means that 4 is multiplied by (giving ) and by 9 (giving 36), and because there is a minus sign inside the brackets, meaning that 9 is taken away from , then 36 is taken away from .
Hence, we find that
Example 3: Using the Distributive Property with Negative Numbers
Using the distributive property, rewrite the expression .
Answer
The expression is a multiplication of two factors: and . The distributive property allows us to rewrite this expression by considering that each term inside the brackets is multiplied by . 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 .
Hence, we have
Example 4: Identifying the Use of the Distributive Property
Which of the following demonstrates the distributive property?
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, , which is the product of and . The distributive property means that the factor is distributed over the two terms inside the brackets; that is,
Carrying out the products in the right-hand side of the equation and writing for , we find indeed that
Hence, the answer is option D.
Example 5: Using the Distributive Property in a Word Problem
Amer had 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 ?
Answer
Amer had dollars to which he added $11, which means that he then had
We are told that this amount doubled after seven months; that is, he then had
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
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.
