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Video: Multiplying Binomials - FOIL

Lucy Murray

Learn how to expand expressions, such as (x + 3)(x - 2). We call this multiplying binomials. We also use the acronym FOIL to remind us to multiply the first, Outer, Inner, and Last terms of the parentheses.

05:36

Video Transcript

Multiply Binomials Using FOIL

This is an example of a binomial. So a monomial is basically one term, whereas a binomial is two terms. So this video, we’re going to focus on multiplying two binomials together. So we’ll multiply all of this binomial by the binomial 𝑥 minus two. Now as I said at the beginning of this video — the title — we’re going to use FOIL. So FOIL means First, Outside, Inside, and Last. And it refers to the order in which we’ll multiply out each of these terms. So first off, we’re looking for the First; so what that means is the first term in each parenthesis. So that would be- so first off, we will multiply 𝑥s.

I like to use these lines as we go through so that when you get to the end of the multiplication you not only know I’ve done the First, I’ve done the Outside, I’ve done the Inside, and I’ve done the Last, but you can see with the lines that you definitely would have done all the calculations you were supposed to do. So first off, we’re saying 𝑥 multiplied by 𝑥. And then we’re going to do the Outside; so that’s the outside terms when we look at the parentheses. So it will be 𝑥 and negative two. Be careful with negatives. Make sure you always multiply the negatives as well; don’t lose them. So now we’ve got the Outside; we’ve got 𝑥 multiplied by negative two as we said. The Inside terms, so as our Outside terms were 𝑥 and negative two, the Inside term will be three and 𝑥. So we’re saying three multiplied by 𝑥. And then finally the Last terms, so the Last terms in each of the parentheses, which we can see is three and negative two, so three multiplied by negative two.

And I’ve written each of our multiplications out here. You won’t always do this, but it’s good for the first time to see where everything is coming from. Looking at the lines now, you can see what I was on about. So you can see all of the multiplications that you’ve done there, and now let’s simplify. So we’ve got 𝑥 multiplied by 𝑥 is 𝑥 squared, 𝑥 multiplied by negative two is negative two 𝑥, three multiplied by 𝑥 is three 𝑥, three multiplied by negative two is negative six.

So now you can see we’ve done all of our multiplications and we’re left with this expression. But we can also collect like terms in this expression. So looking at the middle two terms, we’ve got negative two 𝑥 add three 𝑥 gives us 𝑥. And so we have a final answer of 𝑥 squared plus 𝑥 minus six. We’re gonna look at another example now in a different form and also not with just 𝑥. So given the polynomials 𝐹 of 𝑥 and 𝐺 of 𝑥 below, calculate each of 𝑥 which is equal to the product of 𝐹 of 𝑥 and 𝐺 of 𝑥, where 𝐹 of 𝑥 is 𝑥 squared minus 𝑥 and 𝐺 of 𝑥 is 𝑥 minus three.

So let’s write this out in the form that we just had for the previous example. Although this looks quite different to previous question, it’s pretty much exactly the same. Or at least the method that we have to employ to multiply out these parentheses is exactly the same. So we’re going to use FOIL again. So we’re going to multiply the First terms, which in this case we can see is 𝑥 squared and 𝑥, and then the Outside terms, which we can see in this case will be 𝑥 squared and negative three. Again be careful with the negative. Then the Inside terms, which we can see in this case is negative 𝑥 and 𝑥 and then the Last terms in each parentheses, so that would be negative 𝑥 and negative three.

So 𝑥 squared multiplied by 𝑥 gives us 𝑥 cubed, 𝑥 squared multiplied by negative three gives us negative three 𝑥 squared, negative 𝑥 multiplied by 𝑥 gives us negative 𝑥 squared, and finally negative 𝑥 multiplied by negative three gives us three 𝑥 because a negative multiplied by a negative is a positive.

And again you can see that we’ve not finished and that’s because we have some like terms as our middle two terms again. So we’re going to have to collect them. So first of all, we know that 𝑥 cubed is not like, so we can write that straight down. Then, we’ve got negative three 𝑥 squared minus 𝑥 squared gives us negative four 𝑥 squared. And then finally add three 𝑥. There we have it. We have multiplied out two sets of binomials by using the FOIL method of expansion. And it will always work when multiplying out these binomials. So if you’re happy with it, use it.