Distributive Property: Definition, Use & Examples

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David Liano

After completing this lesson, you will be able to state the distributive property and apply it to various types of problems. You will also be able to accurately perform the mathematical operations that are involved with the distributive property.

We also recommend watching Logarithmic Properties and What Are the Five Main Exponent Properties?

Definition

The distribution property involves the operations of multiplication and addition or multiplication and subtraction. When we use the distributive property, we are multiplying each term inside the parentheses with the term outside of the parentheses. The distributive property, which is displayed below, holds true for all real numbers a, b, and c. Also notice that, if you view the formula in the opposite direction, we are just taking out the common factor of a.

Distributive Property
distributive property

Examples

Let's start with a simple application in arithmetic:

5(3 + 5)

Using the distributive property, we simplify as follows:

5(3) + 5(5) = 15 + 25 = 40

Of course, we would normally add 3 and 5 first and then multiply 5 by 8 to get the same answer. But this basic example allows us to demonstrate the distributive property before we get into more complex problems.

The distributive property can also help when you need to calculate equations using mental math, making some numbers easier to work with. Let's say that you purchased three sandwiches at a local eatery for $5.85 each, including tax; however, you are not sure if you have enough money to pay the check. You can think of $5.85 as $6 - $.15. Then think of the problem as follows:

3(5.85) = 3(6 - .15) = 3(6) - 3(.15) = 18 - .45 = 17.55

Applications in Algebraic Expressions

We can also use the distributive property with variables. Let's simplify the following equation:

7(2x + 7) - 11x

7(2x) + 7(7) - 11x (applying distributive property)

14x + 49 - 11x (simplifying)

3x + 49 (combining like terms)

We also use the distributive property when we multiply two binomials. When we multiply two binomials, we are actually using the distributive property twice. This is commonly referred to as foiling, especially when multiplying the factors of a quadratic equation, as in this example:

(2x + 3)(x - 5)

2x(x - 5) + 3(x - 5) (rewriting expression)

2x(x) - 2x(5) + 3(x) - 3(5) (applying distributive property)

2x^2 - 10x + 3x - 15 (simplifying)

2x^2 - 7x - 15 (combining like terms)

The Distributive Property and Changing Signs

Be careful when you have a negative sign as part of an expression. For instance the expression -(x - 4) really means that we are distributing a (-1) to both the x and the 4 as follows:

-1(x) - (-1)(4) = -x + 4

Make sure that you distribute the negative sign to each term within the parentheses as shown in Figure 1:

Figure 1
negative sign

Let's simplify the following example:

5(4x - 7) - 4(-3x + 8)

5(4x) - 5(7) - 4(-3x) - 4(8) (applying distribute property)

20x - 35 + 12x - 32 (simplifying)

32x - 67 (combining like terms)

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