Special products of binomials
- [Voiceover] Let's see if we can figure out what x plus three times x minus three is, and I encourage you to pause the video and see if you can work this out. Well, one way to tackle it is the way that we've always tackled it when we multiply binomials, is just apply the distributive property twice. So first we can take this entire yellow x plus three and multiply times each of these two terms. So first we can multiply it times this x. So that's going to be x times x plus three. And then we are going to multiply it times, we can say, this negative three. So we could write minus three times, now that's going to be multiplied by x plus three again. And then we apply the distributive property one more time. Where we take this magenta x and we distribute it across this x plus three so x times x is x squared, x times three is three x, and then we do it on this side. Negative three times x is negative three x and negative three times three is negative nine. And what does this simplify to? Well, we're gonna get x squared, and we have three x and minus three x so these two characters cancel out, and we are just left with x squared minus nine. And you might see a little pattern here, notice I added three and then I subtracted three and I got this, I got the x squared and then if you take three and multiply it by negative three, you are going to get a negative nine. And notice, the middle terms canceled out. And one thing you might ask is, well, will that always be the case, if we add a number and we subtract that same number like that? And we could try it out. Let's talk in general terms. So if we, instead of doing x plus three times x minus three, we could write this same thing as, instead of three, let's just say you have x plus a times x minus a. And I encourage you to pause this video and work it all out, just assume a is some number, like three or some other number, and apply the distributive property twice and see what you get. Well, let's work through it. So first we can distribute this yellow x plus a onto the x and the negative a. So x plus a times x, or we could say x times x plus a, that's going to be x times x plus a, and then we're going to have minus a, or this negative a, times x plus a. So minus, and then we're gonna have this minus a times x, plus a. Notice, all I did is I distributed this yellow, I distributed this big chunk of this expression, I just distributed it onto the x and onto this negative a. I'm multiplying it times the x and I'm multiplying it by the negative a. And now we can apply the distributive property again. X times x is x squared, x times a is ax, and then we get negative a times x is negative ax, and then negative a times a, is negative a squared. And notice, regardless of my choice of a, I'm going to have ax and then minus ax. So this is always going to cancel out. It didn't just work for the case when a was three. For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. So this is just going to cancel out, and what are we going to be left with? We are going to be left with x squared minus a squared. And you can view this as a special case. When you have something, x plus something, times x minus that same something, it's going to be x squared minus that something squared. And this is a good one to know in general. And we could use it to quickly figure out the products of other binomials that fit this pattern here. So if I were to say, quick, what is x plus 10, times x minus 10? Well, you could say, all right this fits the pattern, it's x plus a times x minus a, so it's going to be x squared minus a squared. If a is 10, a squared is going to be 100. So you can do it really quick once you recognize the pattern.