Special products of binomials
- [Voiceover] Let's see if we can figure out what x plus seven, let me write that a little bit neater, x plus seven squared is. And I encourage you to pause the video and work through it on your own. Alright, now let's work through this together. So we just have to remember, we're squaring the entire binomial. So this thing is going to be the same thing as: x plus seven times x plus seven. I'm gonna write the second x plus seven in a different color, which is going to be helpful when we actually multiply things out. When we see it like this, then we can multiply these out the way we would multiply any binomials. And I'll first do it the, I guess you can say, the slower way, but the more intuitive way, applying the distributive property twice. And then we'll think about maybe some shortcuts or some patterns we might be able to recognize, especially when we are squaring binomials. So let's start with just applying the distributive property twice. So let's distribute this yellow x plus seven over this magenta x plus seven. So we can multiply it by the x, this magenta x, so it's going to be x, let me do it in that same color. So it's going to be magenta x times x plus seven plus magenta seven times yellow x plus seven. X plus seven, and now we can apply the distributive property again. We can take this magenta x and distribute it over the x plus seven. So x times x is x-squared. X times seven is seven x. And then we can do it again over here. This seven, let me do it in a different color, so this seven times that x is going to be plus another seven x and then the seven times the seven is going to be 49. And we're in the home stretch. We can then simplify it. This is going to be x-squared and then these two middle terms we can add together. Seven x, let me do this in orange, seven x plus seven x is going to be 14 x plus 14 x plus 49. Plus 49. And we're done. Now the key question is do we see some patterns here? Do we see some patterns that we can generalize and that might help us square binomials a little bit faster in the future? Well, when we first looked at just multiplying binomials, we saw a pattern like x plus a times x plus b is going to be equal to x-squared, let me write it this way, is going to be equal to x-squared plus a plus b x plus b-squared. And so, if both a and b are the same thing, we can say that x plus a times x plus a is going to be equal to x-squared, and this is the case when we have a coefficient of one on both of these x's, x-squared's. Now in this case, a and b are both a. So it's going to be a plus a times x, or we can just say plus two a x. Let me be clear what I just did. Instead of writing a plus b, I can just view this as a plus a times x, and then plus a-squared, or that's the same thing as x-squared plus two a x plus a-squared. This is a general way of expressing a squared binomial like this. A squared binomial where the coefficients on both x's are one. We can see that's exactly what we saw over here. In this, in the example we did, seven is our a. So we got x-squared right over there let me circle it. So we have this blue x-squared that corresponds to that over there. And then seven is our a, so two a x , two times seven is 14 x. Notice we have the 14 x right over there. So this 14 x corresponds to two a x, and then finally if a is seven, a-squared is 49. A-squared is 49. So in general if you are squaring a binomial, you could, a fast way of doing it is to do this pattern here, and we can do another example real fast, just to make sure that we've understood things. If I were to tell you what is x minus, I'll throw a negative in here, x minus three squared, I encourage you to pause the video and think about it. Think about expressing this using this pattern. Well this is going to be, in this case our a, we have to be careful, our a is going to be negative three, so that is our a right over there. So this is going to be equal to x-squared. Now two a x, let me do it in the same colors actually, just for fun. So it's going to be x-squared. Now what is two times a times x? A is negative three, so two times a is negative six. So it's going to be negative six x. So, minus six x, that's two times a is the coefficient. And then we have our x there. And then plus a-squared. Well if a is negative three, what is negative three times negative three? It's going to be positive nine. And just like that, when we looked at this pattern, we were able to very quickly figure out what this binomial squared actually is. And I encourage you, you can do it again, with applying the distributive property twice to verify that this is indeed the same thing as x minus three squared.