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Using FOIL to Represent Area

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Today, we're going to talk about using FOIL to represent area. So we're going to start by reviewing how to find the area of a rectangle or a square, and then, we'll do some examples, looking at the area of a rectangular square using FOIL.

So let's start by reviewing how to find the area of a rectangle or square. Let's say I've got a rectangle that has a length of 3 feet and a width of 5 feet. I know that the formula for the area of a rectangle is area is equal to the length times the width. So if my length is 3 feet, I'm going to substitute that in for my length, and my width is 5 feet. Multiplied together, 3 times 5 will give me 15, and my units will be feet squared. So the area of my rectangle is 15 feet squared.

For my second example, let's say I have a rectangular plot of land, and I know that the length of the rectangular plot is x plus 1, and the width is x plus 3. And I want to come up with an expression for the area of the plot of land. So I'm going to start by labeling my picture with the length and the width. So because I have a variable and a number component to the length, I'm going to write them separately. So I'm going to let this piece here be equal to 1 and this piece up here be equal to x, so that my length is x plus 1.

For my width, I'm going to do the same thing. I'm going to let this piece here be equal to 3 and this piece here be equal to x. So now, my width is shown as 3 plus x, or x plus 3. So I'm going to go ahead and determine the area by multiplying my length times the width, and because the length and the width are both binomials-- they have two terms-- I'm going to use FOIL to multiply.

So again, I know my area is equal to the length times the width. My length is x plus 1, and my width is x plus 3. So again, using FOIL to multiply, x times x would give me x squared. x times 3 will give me 3x. 1 times x will give me 1x, and 1 times 3 will give me 3. I can simplify by adding my middle two terms. That will give me an area of x squared plus 4x plus 3.

And I can verify that by looking at my diagram again and figuring out the area of each individual rectangle. So this bottom rectangle has a length of 1 and a width of 3. So area being length times width, I multiply 1 times 3, and I find the area of this rectangle is 3. Similarly, for the rectangle above it, I'll multiply the length of x by width of 3. That will give me 3x for the area of this rectangle. Here, I'm multiplying a width of x by a length of x. x times x will give me x squared. And finally, in this bottom rectangle, I have a width of x and the length of 1, so that's 1x.

And now, we can see that these four areas are the same as the four terms that I had when I used FOIL to multiply my expressions for the length and the width. So by adding up these separate areas, I can see, or verify, that my area for the entire plot of land will be x squared-- 3x plus 1x will give me 4x-- plus 3, which matches the area that I found before.

So for the last example, let's say we've got another rectangle. Although, this time, we want to find the area of just part of it. We know that there is a piece that's missing in the middle of the rectangle that has dimensions 2x by 3x, but we don't know what x is. And we want to find the area of the rectangle outside of that middle rectangle. So we want to find the area shaded in blue.

So I can write the dimensions of my big rectangle. The length of my rectangle will be 13. However, I don't want to find this entire length, because I don't want to include the area here. So I don't want to include the length of 3x, so I'm going to subtract that from 13. So the length of this rectangle where I'm trying to find the area in blue is going to be 13 minus 3x.

Similarly, the width-- I see that the width is 8 inches, but I, again, do not want to include the width here, which is 2x. So I'm going to subtract that from the 8, so my width is going to be 8 minus 2x. So now, to find the area, I can, again, use my formula of length times width. So that's going to be area is equal to 13 minus 3x times 8 minus 2x.

And again, I'm multiplying two binomials, so I'm going to use FOIL. My first two terms multiplied, 13 times 8, is going to give me 104. My outside terms multiplied, 13 and negative 2x, is going to give me negative 26x. My inside two terms, negative 3x and 8, is going to give me negative 24x, and my last two terms, negative 3x and negative 2x, will give me positive 6x squared.

So now, I can combine my middle two terms. Negative 26x minus 24x is going to give me negative 50x. And I can bring down my other two terms. So I've found an expression for my area. However, if this is not written in standard form, it doesn't go in order of terms by highest degree. So I'm going to rewrite this in standard form. So I'll start with 6x squared, then bring down my minus 50x, and end with 104.

So let's go over our key points today. To find the area of a rectangle, you multiply the length by the width. If the length and/or width of a rectangle are represented by a binomial expressions, you multiply the binomials using FOIL to determine an expression for the area.

So I hope that these key points and examples helped you understand a little bit more about using FOIL to represent area. Keep using your notes and keep on practicing, and soon, you'll be a pro. Thanks for watching.