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Multiplying binomials: area model

CCSS.Math:

Video transcript

so I have this big rectangle here that's divided into four smaller rectangles so what I want to do is I want to express the area of this larger rectangle and I want to do it two ways the first way I want to express it as the product of two binomials and then I want to 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 2 so the entire height right over here the entire height right over here is going to be X plus 2 so the height is X plus 2 and what's the width well the width is we go from there to there is X and then from there to there is 3 so the entire width is X plus 3 X plus 3 so just like that I've expressed the area of the entire rectangle and it's a product of two binomials but now let's express it as a trinomial well to do that we can break down the large 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 it's area is x squared let me write that that's x squared what's the area of this yellow rectangle well its height is X same height is right over here its height is X and its width is 3 so it's going to be x times 3 or 3 X I'll have an area of 3 X so that area is 3 X so if we're summing up the area of the entire thing this would be plus 3 X 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 2 and the width is X so multiplying height times width is going to be 2 times X and we can just add that plus 2 times X and then finally this little gray box here its height is 2 we see that right over there it's height is 2 and its width is 3 we see it right over there so it has an area of 6 2 times 3 so plus 6 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 I have three X's and I had two x's to that I'm going to have five X's so this entire thing simplifies to x squared x squared plus 5x plus 6 plus 6 so this and this are two ways of expressing the area so they're going to be equal and that makes sense because if you multiply it out these binomials and simplified you would get this trinomial we can do that really fast you multiply the x times the X X let me do it in the same colors you multiply you multiply the x times the X you get the x squared you multiply this x times the 3 you get your 3x you multiply you multiply the 2 times the X you get your 2x and then you multiply the 2 times the 3 and you get your 6 so what this with this because you can see this area model does 4 is is it hopefully makes a visual representation of why it makes sense to multiply binomials the way we do and we in other videos we talked about it is applying the distribute the distribution property twice but this gives you a more visual representation for why it actually makes sense