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*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?

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*.

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

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

7(2*x* + 7) - 11*x*

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

14*x* + 49 - 11*x* (simplifying)

3*x* + 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:

(2*x* + 3)(*x* - 5)

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

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

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

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

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:

Let's simplify the following example:

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

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

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

32*x* - 67 (combining like terms)

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When you take a course in geometry, don't think that you can forget about algebra for a while - the two mathematical topics are very much linked. Algebra is a very important tool when solving geometric problems. Let's consider the following problem:

The length of a rectangle is 5 more than its width. The total area of the rectangle is 84 square units. What is the width and length of the rectangle?

Let's call the width *x*. That means that the length is *x* + 5. It probably helps to draw a picture, as shown in Figure 2.

We know that the area of a rectangle is width multiplied by length, so we can write the following equation and solve for *x*:

*x*(*x* + 5) = 84

*x*^2 + 5*x* = 84 (applying distributive property)

*x*^2 + 5*x* - 84 = 0 (writing as a quadratic equation)

(*x* + 12)(*x* - 7) = 0 (factoring)

The possible solutions for *x* are -12 and 7. We know that the side of a rectangle cannot be negative, so *x* must equal 7. The width is *x* = 7 and the length is (7) + 5 = 12.

The distribute property can be extended for additional terms. For instance *a*(*x* + *y* + *z*) = *ax* + *ay* + *az*.

We can also use the converse of the distributive property, which is factoring. If we wanted to factor *x*^2 - 37*x*, we would get *x*(*x* - 37).

The distributive property allows us to multiply one factor with many different factors that are being added and/or subtracted together. The property often makes problems solvable mentally or at least easier to solve.

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