Hi, and welcome. This is Anthony Varela. And today we're going to use FOIL to represent area. So we're going to review our concept of area, specifically with rectangles.
And we're going to represent our dimensions with binomials. So this is going to involve binomial multiplication. And we use FOIL to multiply binomials.
And then we're going to go through examples where we're adding areas and then also subtracting areas. So let's review area of rectangles. Well, we just multiply the two dimensions. So we call our dimensions length and width. So the area of a rectangle is the length times the width.
So for example, here we have this rectangle that measures three feet by 4 feet. So we multiply three and four ticket 12. And we also multiply our units feet by feet to get squared feet as our measurement for area.
Well, we're going to take a look at binomial multiplication with area. So our dimensions are going to be expressed as binomials.
So for example, we might have this green rectangle that is 3 by 4. But then there's this border that goes around. So our dimensions are actually going to be binomial. So we're going to see how we can represent our area that way.
So let's start with our first example. Here we have a floor plan of a house or an apartment. And there are four rooms on this floor-- the dining room, the bathroom, the living room, and the kitchen.
Now here's what we know about the dimensions of these rooms. So our living room is 12 feet by 15 feet, and our bathroom is a square room. We don't know the dimensions. So we're going to describe them as x feet by x feet.
So we can actually put the pieces together then and express the area of the different rooms. So this creates four rectangles within our larger rectangle. And we can say then that the bathroom has an area of x squared squared feet. And the living room then, if we multiply 12 and 15, we get 180 squared feet as the area of the living room.
Now what about our dining room? Well, its dimensions are 15 feet by x feet. So the area is 15x squared feet.
And how about the kitchen? Well, its dimensions are x feet by 12 feet. So the area of the kitchen is 12x squared feet.
Now let's compare this then to the dimensions of the entire floor. So it has dimensions of x plus 15, here, and x plus 12 here. So we can find the area of the entire floor by multiplying these two binomials, x plus 15 times x plus 12.
And when we FOIL, we multiply the first terms, the outside terms, the inside terms, and then the last terms. And we can always combine these two x terms to simplify our expression.
Well, taking a look at x plus 15 times x plus 12, if we were to FOIL that, what we would get is x squared plus 15x plus 12x plus 180. And that's exactly what we see here, x squared plus 15x plus 12x plus 180. And we can simplify this to x squared plus 27x plus 180 when we add 15 and 12-- 15x and 12x.
So now let's go through an example with subtracting area with binomial multiplication. So our situation is a pool out in the backyard. And it has a walkway that goes around the pool. And with the walkway included, the dimensions are 8 meters by 5 meters. And the walkway is made up of square blocks that go all the way around.
So we can say that the dimensions of each square block is x meters by x meters. So how can we describe then the area of this top surface then to the pool? Well, taking a look at the width of this figure, we have a total of 5 meters. But we know that we have x meters here. And we know that we have x meters on the other side as well.
So we're going to start with 5 meters, but we'll be subtracting something. And we'll be subtracting x twice. So we can say then that 5 minus 2x is one dimension.
How about the other dimension? Well, it's 8 meters to start with. But once again, we're subtracting x here and we're subtracting x here. So our length then is 8 minus 2x again.
So notice here then our binomials involve a subtraction sign somewhere. So it could be x minus a or a minus x. There's going to be subtraction in there.
And what we can do then to find the area of this top surface is multiply these two dimensions. So multiplying negative 2x plus 8 and negative 2x plus 5, notice what I've done is just put my x term first, carrying the negative with it, just so that it looks more like something I'm used to when I'm FOILing.
And so when I multiply this out, I'll get positive 4x squared. Then I'll subtract 10x, subtract 16x, and then I'll add 40. And when I combined my x terms, I get 4x squared minus 26x plus 40. And my units are in squared meters.
So let's review using FOIL to represent area. Well, we talked about the area of a rectangle being length times width. And if our length and width are expressed as binomials, we can use FOIL to multiply those two dimensions.
We looked in an example where we were adding individual areas. And so our binomials had a plus sign in there. And then we also looked at an example where we were subtracting area, an example with the pool and the walkway. So our binomials had negatives or a subtraction sign in there.
So thanks for watching this tutorial on using FOIL to represent area. Hope to see you next time.