Area model for multiplying polynomials with negative terms

In previous videos, we've already looked at using area models to think about multiplying expressions, like multiplying x plus seven times x plus three. In those videos, we saw that we could think about it as finding the area of a rectangle, where we could break up the length of the rectangle as part of the length has length x, and then the rest of it has length seven. So this would be seven here, and then the total length of this side would be x plus seven, and then the total length of this side would be x plus, and then you have three right over here.

What area models did is they helped us visualize why we multiply the different terms or how we multiply the different terms because if we're looking for the entire area, the entire area is going to be (x plus 7) times (x plus 3).

And then, of course, we can break that down into these sub-rectangles. This rectangle, and this is actually going to be a square, would have an area of x squared. This one over here will have an area of 7x (7 times x). This one over here will have an area of 3x, and then this one over here will have an area of 3 times 7 or 21.

So we can figure out that the ultimate product here is going to be x squared plus 7x plus 3x plus 21. That's going to be the area of the entire rectangle. Of course, we could add the 7x to the 3x to get to 10x. But some of you might be wondering, well, this is all nice when I have plus 7 and plus three. I can think about positive lengths; I can think about positive areas.

But what if it wasn't that way? What if we were dealing with negatives instead? For example, if we now try to do the same thing, we could say, all right, this top length right over here would be x minus seven. So let's just keep going with it, and let's call this length negative seven up here, so it has a negative 7 length.

And we're not necessarily used to thinking about lengths as negative. Let's just go with it. And then the height right over here would be x minus 3. So we could write an x there for that part of the height, and for this part of the height, we could put a negative 3.

So let's see if we kept going with what we did last time. The area here would be x squared. The area here would be negative 7 times x, so that would be negative 7x. This green area would be negative 3x, and then this orange area would be negative 3 times negative 7, which is positive 21.

And then we would say that the entire product is x squared minus 7x minus 3x plus 21. And we could, of course, add these two together to get negative 10x. But does this make sense? Well, one way to think about it is that a negative area is an area that you would take away from the total area.

So if x happens to be a positive number here, then this pink area would be negative, and so we would take it away from the whole. And that's exactly what is happening in this expression. It's worth mentioning that even before, when this wasn't a negative 7, when it was a positive 7 and this was a positive 7x, it's completely possible that x is negative, in which case you would have had a negative area anyway.

But to appreciate that this will all work out, even with negative numbers, I'll give an example if x were equal to 10. That will help us make sense of things. So if x were equal to 10, we'd get an area model that looks like this: we're having 10 minus 7, so I'll put minus 7 right over here, times 10 minus 3.

Now you can figure out in your heads what's that going to be. 10 minus 7 is 3; 10 minus 3 is 7, so this should all add up to positive 21. Let's make sure that's actually occurring. So this blue area is going to be 10 times 10, which is 100.

This pink area now is 10 times negative 7, so it's negative 70. So we're going to take it away from the total area. This green area is negative 3 times 10, so that's negative 30, and then, negative 3 times negative 7, this orange area is positive 21.

Does that all work out? So let's see. If we take this positive area, 100 minus 70 minus 30, and then add 21, 100 minus 70 is going to be 30; minus 30 again is zero, and then you just have 21 left over, which is exactly what you would expect.

You could actually move this pink area over and subtract it from this blue area, and then you could take this green area and then you could turn it vertical, and then that would subtract out the rest of the blue area. And then all you would have left is this orange area.

So hopefully this helps you appreciate that area models for multiplying expressions also works if the terms are negative. And also a reminder: when we just had x's here, they could have been negative to begin with.