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Multiplying Polynomials using Area Models - Problem 2
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A lot of students like FOIL because it’s easy to remember, but one of the bummers with FOIL is that it only works for multiplying 2 binomials. It wouldn’t work in this problem because I have a binomial and a trinomial. So I could do a big distributing process and then combine like terms or I could use the area model. Here is what that would look like.

I’m going to draw a rectangle that’s 2 by 3 because I have a binomial with 2 terms and a trinomial with 3 terms. It doesn’t matter if you draw it this way where you have the trinomial down the side, binomial across the top, or if you draw that shape on its side which I’m going to do. Either way is fine. I’m going to draw it like this just because it makes more sense in my head that way, but it’s totally up to you if you want to draw this shape turned up, up and downy.

Okay so my binomial is 2x-4, that’s going to go here. Across the top I’m going to have my big trinomial, -3x² be careful don’t lose that negative sign plus 5x take away 6. Let’s go through and fill in the boxes and then combine like terms. -3 times 2 is -6 and then x times x² is x to the third. +5 times 2 is 10 and then x times x is x², -6 times +2 is -12, don’t forget the x. Go through and multiply, same process -20x and then +24 at the end.

Okay and the next thing I need is to write my final answer by combining like terms. One thing you might notice when you guys do lots and lots of these problems, is that your like terms show up diagonally like this. If that does not make sense to you don’t worry about it, but some people can see that the x²' and the regular Xs show up in these diagonals.

So my final answer is going to be -6x to the third, here are my x²', I have +10 plus 12 more so that’s 22x². -12 take away 20 is -32xs and then plus 24 at the end. That’s my final answer. Not too bad when you use the area model, you can be sure that all of your products are being accounted for. You’re multiplying both of these binomial terms by each of these trinomial terms and then combining the like terms. Just be really careful, again don’t lose the negative signs that -6 that -3 that -4 are super, super important and if you’re making mistakes in your homework chances are you’re just losing a negative.

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