Multiplying Polynomials using Area Models - Problem 1

Here I’m given 2 binomials multiplied together. So in order to do this problem, I could use FOIL, but this problem asks me use an area model.

So I’m going to draw a rectangle that’s 2 by 2 to represent binomial times binomial. Then across the top I’m going to write my first one 3x plus 1, down the side I’m going to write my second one x take away 2, and this is a little tricky because some of you guys are thinking you can’t have a negative area and that is a little weird because when we fill in these boxes, you’re going to see we’re going to get a negative quantity in one of them and that’s weird like area can’t be negative. That’s why this concept has a little bit of a trick in your brain. You just got to make sure you’re keeping track of the positive and negative signs and trust that the process works for multiplying binomials.

3x plus 1 I wrote across the top x minus 2 and down the side, so here 3x times x inside this box I’m going to have 3x². Here I’m going to do 1 times x to give me this plain old 1 x. Here 3x times -2 is -6x and then here I’ll have -2. It’s kind of weird because they’re negative areas. I’m just using this idea to help me find the product.

Last thing is to combine like terms, 3x², here I have 6 and -6x, excuse me I have x and -6x, those can be combined to -5x take away 2, that’s my final answer that I would have gotten through Foiling, I just wrote out in a different way. So this Area Model can be really useful because it helps you organize each of your different products for multiplying in these polynomials.