What is the Distributive Property?

The distributive property is expressed in math terms as the following equation: a(b + c) = ab + ac. You can read this as the sum of a(b + c) is equal to the sum of a times b and a times c. When you’re looking at an equation like this, you can see that the multiplication part distributes evenly to all the numbers within the parentheses. It would be incorrect to multiply ab and just add c, or to multiply ac and add b. The distributive property reminds us that everything within the parentheses needs to be multiplied by the outside number.

Students may first learn the distributive property when they are learning order of operations. This is the concept that in problems where there are different mathematical operations, such as multiple, addition, subtraction, parenthesis, you have to work in a certain order to get the right answer. This order is parentheses, exponents, multiplication and division. and addition and subtraction, which may be abbreviated to PEMDAS.

When you have a math problem that uses parentheses you need to solve what’s in the parenthesis first, before you can move on to solving other problems. If the math problem simply has known numbers, it’s fairly easy to solve. 2(10+5) becomes 2(15) or is also equal under the distributive property to 2(10) + 2(5). What gets more complicated is when you are working with variables (a, b, x, y, and so on) in algebra, and when these variables cannot be combined together.

Consider the equation 9(10a + 2). If we don’t know what the variable a stands for, we can’t add 10a + 2, but using the distributive property still allows us to simply this expression because we know this equation is equal to 9(10a) + 9(2). In order to simply the expression we can take each part separately and multiply it to 9, and we get 90a + 18.

Another way to use the distributive property is if you want to figure out the similarities in an equation. In the example 90a + 18, although the terms are not like, they have something in common. You can work backward to take out the factor of 9 and put the unlike terms in parentheses. Thus 90a + 18 can equal 9(a +2). We have removed the element that is common to these terms, the common factor of 9.

Why on earth would you want to work the distributive property backward? Say you have an equation that 4a + 4= 8. Using the distributive property before we get to subtracting terms to solve for a, can simplify the work. You can divide the entire equation on both sides by 4, giving us the answer a + 1 =2. From there it’s easy to determine that a =1. Sometimes it makes sense to reduce unlike terms by their common factor to more easily solve an equation.