What is the Distributive Property?

As you prepare to take the SAT or ACT, you should know that there will inevitably be some basic algebra in your near future. Regardless of which of the two standardized tests you take, you will see concepts such as manipulating linear equations, understanding and transforming graphs, and solving for multiple variables in an equation. So as you’re preparing for the exam, it’s best to review these concepts.

One of the key concepts that you will need to simplify, manipulate, and interpret any linear equation is the distributive property. You may have dealt with it in a previous algebra class or heard of it as the term FOIL. If you master this concept, you will be well on your way to conquering many more problems on the SAT or ACT. Let’s review what exactly the distributive property is, how it’ll be used on the SAT, and do a few practice questions with it.

What is the Distributive Property?

At the most basic level, the Distributive Property is a rule that tells you how to handle the multiplication of numbers or variables inside parentheses by a separate numbers/variables outside of the parentheses. Let’s look at an example:

\(4(8+3)\)

As you can see, \(8\) and \(3\) are inside the parentheses, and they are meant to be added. However, there is a \(4\) outside of the parentheses. The presence of the \(4\) indicates that you need to multiply whatever is inside the parentheses by \(4\). This would be simple if what was inside the parentheses was a single number, like \(3\) or \(10.5\). Then you could just multiply \(4\) by that number. Instead, we have to to multiply \(4\) by a separate equation. That is not so intuitive.

The Distributive Property tells us that, in order to solve this problem, you need to distribute the \(4\) to each variable in the equation in the parentheses. Then, you can just complete the equation. Here is what that looks like:

\(4(8+3)\)

\(=4(8)+4(3)\)

\(=32+12\)

\(=44\)

Notice how we didn’t alter the equation inside the parentheses at all. All we did was multiple each number inside the parentheses by four. This is what distributing the variable means.

Now, when you’re just dealing with numbers instead of variables like \(x\) and \(y\), this seems like a redundant process. Indeed, it would be much easier to just do this:

\(4(8+3)\)

\(=4(11)\)

\(=44\)

In this case, we got the same answer, and there were fewer steps involved. However, when you are dealing with a combination of variables and numbers, or just variables, you can’t use the above method and the Distributive Property becomes much more important. Take, for example:

\(4(x+3)\)

You can’t use the simpler method. What is \(x+3\)? We don’t know \(x\), so it’s impossible to know. Let’s try using the Distributive Property:

\(4(x+3)\)

\(=4(x)+4(3)\)

\(=4x+12\)

Now, we have an answer that is simplified as much as possible without knowing the values of any variables. This is the distributive property at work!

It’s important to note that you don’t need real numbers at any point in this equation to use the Distributive Property. In fact, you can have all variables and still get a simplified answer:

\(=x(w+y+z)\)

\(=xw+xy+xz\)

Thus, the Distributive Property is a useful tool that you can use whenever you need to manipulate and simplify equations.

One last note: The Distributive Property goes hand in hand with the FOIL method, which involves multiplying two sets of equations together. For instance, you need to use the FOIL method to simplify this expression:

\((x+3)(x+5)\)

Which one do you distribute, and how do you distribute it to the other equation? You use the FOIL method, and in fact, the answer is in the name:

F- multiply the first numbers/variables in each equation

\(x\:\cdot\:x = x^2\)

O- multiply the two numbers/variables on the outside of the expression

\(x\:\cdot\:5 = 5x\)

I- multiply the two numbers/variables on the inside of the expression

\(3\:\cdot\:x = 3x\)

L- multiply the last numbers/variables in each equation

\(3\:\cdot\:5 = 15\)

Once you’ve followed the steps of the FOIL method, add all of your results together to get the simplified expression:

\(x^2+5x+3x+15\)

\(=x^2+8x+15\)

That’s all there is to it! As you can see, the FOIL method is just a way of teaching you how to extend the distributive property to more complicated multiplication tasks.