Distributive Property

Definition

The distributive property is the ability of one operation to "distribute" over another operation contained inside a set of parenthesis. Most commonly, this refers to the property of multiplication distributing over addition or subtraction, such that \(x(a+b) = xa + xb\).

When we say that multiplication distributes over addition, it means we can distribute the multiplicative factor outside the set of parenthesis to each item inside, and then add the results. For example, \(4(3+7)\) is equivalent to \(4*3 + 4*7\) because the multiplication by four was distributed across the addition inside the parenthesis.

Not every operation is distributive. For example, division is not distributive over addition. If we are given \(\frac{20}{(3+7)}\) the correct result is 2, but distributing would give you \(\frac{20}{3} + \frac{20}{7}\), which is about 10 and is very incorrect!

Application

The distributive property is actually a very simple concept to learn and apply. It will allow you to simplify something like \(3(6x + 4)\), where you have a number being multiplied by a set of parenthesis. Let's start with a simple problem:

$$6 (4 + 2) $$

Based on the order of operations, you know that anything inside parenthesis should be done first. Adding \(4 + 2\) is simple enough, resulting in this:

$$ 6(6) $$

When you see a number next to parenthesis like this, it implicitly means multiplication, so what we really have here is this (remember that \(*\) means multiplication):

$$ 6 * 6 = 36 $$

That was easy enough, but we can also solve this problem and get the same answer using the distributive property of multiplication over addition.

$$ 6(4 + 2) $$

Now distribute the 6 across the parenthesis to the two terms inside:

$$ (6 * 4) + (6 * 2) $$ $$ 24 + 12 $$ $$ = 36 $$

Now try simplifying this expression:

$$ -2(4y - 8) $$

This is no more difficult to simplify than the last one. Just distribute the \(-2\) to the terms inside, which are the \(4y\) and the \(-8\):

$$ (-2 * 4y) + (-2 * -8) $$ $$ -8y + 16 $$ $$ 16 - 8y $$

And that's all there is to it. Once you get the hang of things it will be second-nature to you. You simply have to remember that only certain operations are distributive: multiplication distributes over addition and subtraction, but division does not! You are welcome to continue browsing our site now, or you can read another lesson on the distributive property from Wyzant or this lesson from PurpleMath.