Alex’s Guide to the Distributive Property
This is one of the most commonly used tools in algebra, and it’s of paramount importance that you are
comfortable with it. If you look back at the Basics of Algebra guide, it lists distribution as one of the algebraic
properties. Distribution means “multiplying something by everything else.” Later in the year, we will learn the
FOIL method, which is essentially “double-distribution.” For the sake of thoroughness, I have included an
example of FOILing below too.
Using the Distributive Property
A basic example of the distributive property is:
2(3 + 4)
There are two ways to evaluate this:
Method 1: Do the addition in the parentheses first.
Method 2: Distribute the 2 first.
2(7)
14
2(3) + 2(4)
14
As you can see, either method gives the same result. So why, you may ask, would you want to do the more
difficult-seeming distribution? The answer is: you have to when you’re dealing with variables, instead of just
numbers. Make sure you distribute what’s outside the parentheses to everything that’s inside the parentheses.
Here’s an example:
2(𝑥 + 4)
Since there is a variable, we have to use the distributive property to simplify this expression:
2(𝑥) + 2(4)
And simplifying again gives us:
2𝑥 + 8
FOILing
This is what you do when you have two or more things multiplied by two or more other things. Again, the
mantra is, “Multiply everything in one group by everything in the other.” The easiest way to think of this is the
acronym FOIL, which stands for First-Outer-Inner-Last:
(𝑥 + 2)(𝑥 + 6)
𝐹𝑖𝑟𝑠𝑡: 𝑥 ∙ 𝑥
𝑂𝑢𝑡𝑒𝑟: 𝑥 ∙ 6
𝐼𝑛𝑛𝑒𝑟: 2 ∙ 𝑥
So, you end up with:
𝑥 2 + 6𝑥 + 2𝑥 + 12
This can be simplified, by combining the like middle terms, to:
𝑥 2 + 8𝑥 + 12
𝐿𝑎𝑠𝑡: 2 ∙ 6