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Multiplying binomials: area model

CCSS Math: HSA.APR.A.1

Video transcript

- [Voiceover] So I have this big rectangle here that's divided into four smaller rectangles, and what I wanna do is I wanna express the area of this larger rectangle and I wanna do it two ways. The first way I wanna express it as the product of two binomials and then I wanna express it as a trinomial, so let's think about this a little bit. So one way to say well look, the height of this larger rectangle from here to here, we see that that distance is x, and then from here to here it's two, so the entire height right over here, the entire height right over here is going to be x plus two. So the height is x plus two, and what's the width? Well the width is, we go from there to there is x, and then from there to there is three, so the entire width is x plus three. x plus three. So just like that, I've expressed the area of the entire rectangle, and it's the product of two binomials. But now let's express it as a trinomial. Well to do that, we can break down the larger area into the areas of each of these smaller rectangles. So what's the area of this purple rectangle right over here? Well the purple rectangle, its height is x and its width is x, so its area is x squared. Let me write that, that's x squared. What's the area of this yellow rectangle? Well it's height is x, same height as right over here, its height is x and its width is three, so it's gonna be x times three, or 3x. It'll have an area of 3x. So that area is 3x, if we're summing up the area of the entire thing, this would be plus 3x. So this expression right over here, that's the area of this purple region, plus the area of this yellow region, and then we can move on to this green region. What's the area going to be here? Well the height is two and the width is x, so multiply height times width is gonna be two times x, and we can just add that, plus two times x, and then finally this little grey box here. Its height is two, we see that right over there, its height is two, and its width is three, we see it right over there, so it has an area of six. Two times three. So plus six, and you might say well this isn't a trinomial, this has four terms right over here, but you might notice that we can add, that we can add these middle two terms, 3x plus 2x, if I have three x's and I add two x's to that, I'm gonna have five x's. So this entire thing simplifies to x squared, x squared plus 5x plus six. Plus six. So this and this are two ways of expressing the area, so they're going to be equal, and that makes sense 'cause if you multiply it out, these binomials, and simplify it, you would get this trinomial, and we can do that really fast. You multiply the x times the x, actually let me do that in the same colors. You multiply the x times the x, you get the x squared. You multiply this x times the three, you get your 3x. You multiply the two times the x, you get your 2x. And then you multiply the two times the three and you get your six. So what this, I guess you can say this area model does for us is it hopefully makes a visual representation of why it makes sense to multiply binomials the way we do, and in other videos we talk about it as applying the distribution property twice, but this gives you a more visual representation for why it actually makes sense.