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Special Products of Binomials

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Hi, this is Anthony Varela. And today, we're going to go through some special products of binomial. So we're going to start by reviewing FOIL. That's the process we use to multiple any two binomials. But then we're going to introduce some special products, perfect square trinomials and difference of squares.

So first, recall that FOIL stands for First, Outside, Inside, and Last. So if you'd like to multiply these two binomials, we multiply their first two terms to get 8x squared. Then we multiply the two outside terms to get 4x. Then we multiply the two inside terms to get negative 12x. And then we multiply the last two terms to get negative 6.

Now, this always results then in two x terms that we can combine because they are like terms. So this simplifies to 8x squared minus 8x minus 6. So that's our general process for multiplying any two binomials. But there are some special products that I'd like to talk about.

The first one is a perfect square trinomial. There are actually two different types. So here, we have x plus a times x plus a. So it's the same factor that we're multiplying to itself. And when we FOIL this out, we get x squared plus a x plus ax again and then plus a squared. So we can combine then those two like terms, and we see x squared plus 2ax plus a squared. And now this is a perfect square trinomial because we have three terms here. So something with three terms is a trinomial.

And more specifically, we can also call this the square of a binomial sum because we can write x plus a times x plus a as x plus a squared. So it's a binomial that we're squaring. And noticed here that we have 2ax. So our coefficient in front of our x term is 2a. And then our constant term is a squared.

So if we have something like x plus 4 quantity squared, what I could do is I could write it out as x plus 4 times x plus 4 and go ahead and FOIL. But I could recognize this as a perfect square trinomial, square of a binomial sum, and I could just write this then as x squared plus 8x plus 16. And I knew that right off the bat because I take this value here, double it, and I get my coefficient to the x term, square it, and I get my constant term.

Now, another special product is x minus a times x minus a. Now, if we were to FOIL this out, we'd get x squared minus ax minus ax and plus a squared again because a negative times a negative is a positive. So it's very similar to our one before, but this is then called the square of a binomial difference because we have x minus a instead of x plus a. But notice, it's still one of our perfect square trinomials because here we have 1, 2, 3 terms.

So if we have something like x minus 3 quantity squared, once again I could write this out and FOIL it, or I could recognize this as one of our special products. And I can know right off the bat that this is x squared minus 6x plus 9, once again, taking our negative 3 and doubling-- that's the coefficient of the x term-- but squaring it, and that is the constant term.

So a perfect square trinomial is a polynomial with three terms which can be simplified as a binomial squared, x plus a quantity squared. That would be specifically our square of a binomial sum. If this was x minus a quantity squared, that would be r square of a binomial difference.

I'd also like to talk about difference of squares. So this would be if we have x plus a multiplied by x minus a. So FOILing this, we get x squared. Then we have minus ax plus ax and minus a squared. And notice here, because we have a minus ax and a plus ax, our x terms actually cancel, and we have x squared minus a squared.

So that's why we call it a difference of squares. Because we have one square, another square; we're taking the difference of the two. And this would also work if we just swap these around. X minus a times x plus a will give you the same result.

So if I have x plus 8 and I multiply that by x minus 8, I can recognize this as a difference of squares. And I would know then that we would just have x squared minus 8 squared, which is 64. So I don't have to go through the process of expanding and then FOILing to write this out. So a difference of squares are two squared terms separated by subtraction, x squared minus a squared. It can also be expressed as x plus a times x minus a. So recognizing a special product will help you either expand or factor an expression a little bit quicker than otherwise.

So for example, if we have x squared minus 12x plus 36, I could recognize this as a square of a binomial difference. I have x squared minus 2ax plus a squared. So I can then factor this then as x minus 6 quantity squared. Let's try another one. If we have x squared minus 144, I can recognize this as a difference of squares. 144 is 12 squared. So I know that this would be a difference of squares, x plus 12 times x minus 12.

So let's review our special products of binomials. We talked about perfect square trinomials. We have the square of a binomial sum, x plus a quantity squared, and our square of a binomial difference, x minus a quantity squared. We also talked about the difference of squares, x plus a times x minus a. So Thanks for watching this tutorial on special products of binomials. Hope to see you next time.