Algebraic Expressions: The FOIL Method

What is the FOIL Method in algebra? This video will teach you what the acronym stands for, as well as how to use this valuable tool.

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Transcript: The FOIL Method

Two Expressions That Each Include Addition and Subtraction

Now when can talk about the FOIL method. In the previous lessons, we multiplied expressions in situations in which no more than one of the factors had addition or subtraction. When we multiply two expressions, each of which involves addition or subtraction, how to distribute becomes a little bit trickier.

So think about this. Here we have two binomials.

That is to say, two factors each of which involves the addition of two terms. So how do we distribute? Well, technically, we would have to distribute one factor. So let’s say that second factor r + x, that whole thing is a factor, and we have to distribute it across the first edition. And so we get p times that factor plus q times that factor.


And then once we get to that point, we’d have to distribute in each one of those terms. Distribute the p in the first term, distribute the q in the second term. And so that is technically what we would have to do to distribute. You can think of the process using the Distributive Law in that way, but there’s a very convenient shortcut, summarized by the mnemonic FOIL–the word FOIL.

What Is the FOIL Method?

So what does FOIL stand for? Well, FOIL stands for first. So by first we mean the product of the first two terms inside the parentheses p and r. Each one of those is first in its own parenthesis. And so the product of the first terms, that’s one of the terms.

Then we look at the outer pair. That is to say the very first in the first parenthesis and the very last in the last parenthesis. Those are the outer ones. Then I, those, of course, are the inner ones. The last in the first parenthesis and the first in the last parenthesis, q and r.

And then finally, we look at the two last terms, the last in the first parenthesis and the last in the last. So those are the four pairs we’re gonna look at, summarized by:

  • First
  • Outer
  • Inner
  • Last
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Image by valkoinen

The product of the binomials is the sum of those four individual products. So in other words, if we add first plus outer plus inner plus last, that is the product of the two binomials.

Example One

Here’s an example of the use of FOIL. So let’s go through this very slowly. Here we have two binomials. The first product we’re gonna take, the F, that’s the first term. So that is the 2x from the first parenthesis and the x from the center, the second parenthesis, so (2x) (x) = 2x squared.

Now we’re gonna look at the outer products, that would be the 2x at the beginning and the 2y at the end, that product is 4xy. Now we’ll look at the inner products, that would be the y times the x right in the middle, (y)(x) = xy. And then we’ll look at the last. So that would be (y)(2y), which would give 2y squared.

Now that we have those four individual products, we add them. And, of course, when we add them, we have like terms, you often get like terms in foiling. And so we simplify by combining the like terms, and that is the product.

Example Two

Here’s another example of foiling. So again, two binomials, this time involving subtraction.

So the first would be 2x times 3x, that would be 6x squared. The outer would be 2x times -1, we have to remember to include the negative, so that would be -2x. Then -5 times 3x, that would be the inner, that would give us -15x. And then the final term, -5 times -1 gives us +5.

And again, combine the like terms. Here’s another one with some higher powers, this is good practice for that power rule. When we multiply here, we get x to the 4th times x to the 5th. And as we talked about in the previous video, what happens when we multiply powers as we add the exponents.

So 4 + 5 is 9. So x to the 4th times x to the 5th is x to the 9th. And if that’s something that’s unfamiliar to you, I would suggest go back and watch the previous video or go ahead and watch the powers and roots videos where this is explained in much more detail. The outer terms, x to the 4th times x squared, 4 + 2 is 6, so that would x to the 6th.

The inner terms, x times x to the 5th. 1 + 5 is 6, so that’s also x to the 6th. And then the last terms, x times x squared would be x cubed. And again, finally, add the like terms.

Practice Problems

Here are some practice problems. Pause the video here and work these out on your own.

So the first one we FOIL out, we get those terms and they simplify. The second one we get these terms and they simplify. The third one we get these terms. In this context, we can also discuss a very common algebraic mistake pattern. And just as it’s important to know the correct things to do, it’s also incredibly important to understand the common mistake patterns.

Because these common mistake patterns are very often tempting in correct answer choices that many people pick because they make these mistakes. So here’s the common mistake. If we have a binomial squared, so many people are gonna be tempted to say (a + b) quantity squared = a squared + b squared. They’re gonna be tempted to distribute that exponent, and that is 100% incorrect.

It is absolutely illegal to distribute an exponent across addition or subtraction. We can distribute multiplication across division and subtraction, that’s the distributive law. We cannot distribute an exponent. Instead, squaring anything means multiplying it by itself. Thus, squaring a binomial means multiplying the binomial by itself, and we would FOIL in that process.

So if we do this properly, (a+b) squared, well anything squared is that thing times itself, so it would be (a+b)(a+b), that’s what it means to square something. Well now we have a product of binomials, so now we would FOIL. We get those terms, we’d combine, and we’d get that. So this is very different from just a squared + b squared.

We get what’s called a cross-term, 2ab–the term where the two variables are multiplied together. And in fact this is a very important pattern. This rule (a + b) squared = a squared + 2ab + b squared is called the sum of a square. And it’s very good to be familiar with this pattern, it’s very good to FOIL this out and practice this until you really know this pattern inside out.

A similar rule, the square of a difference can also be found by foiling. This is (a- b) squared = a squared- 2ab + b squared. These two formulas, the square of a sum and the square of a difference are good to remember because they’re two of the most important patterns in all of algebra. And again, Ill make the distinction here, don’t just blindly memorize them, practice the foiling, FOIL them out.

And that way you’ll really own these formulas because you’ll understand where they come from. In summary, we discussed the FOIL method for multiplying two binomials. And we discussed two important formulas, the square of a sum and the square of a difference.

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  • Mike MᶜGarry

    Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.

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