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We're told that the glass portion of the window shown

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below-- that's the window-- has a height to width the

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ratio of 3:2.

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So the height of the glass portion can be represented as

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3x and the width of the glass portion can be

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represented as 2x.

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The wood trim adds 7 inches to the total width and 8 inches

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to the total height.

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So let me write that down.

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So 8 in height, we're doing it in yellow.

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So 8 inches to the total height, I'll do it in yellow,

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and 7 inches to the total width, I'll do in this

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purplish color, right?

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We're doing the width in purple.

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Now what they say is write a polynomial expression that

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represents the total area of the window, including the

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glass and wood.

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Simplify the expression.

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Now, the way they've described it, they're telling us the

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ratio of 3:2 for the glass portion.

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I think when they're talking about the glass portion,

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they're talking about this entire portion right here.

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Even though it's not all glass, you have that thing in

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the middle, you have this divider right over here,

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that's not glass, but I think they're including that as the

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glass portion, because they're not giving us any

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measurements for that.

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So the height of the glass portion can be

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represented as 3x.

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So this height right here of just the glass portion is 3x.

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You can't see that.

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Let me do it in a better color.

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This portion right here is 3x, just this distance right here.

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That is the height of the glass portion.

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And then they tell us the wood trim adds 8 inches to the

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total height.

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So if you take that 3x and then you add that plus that,

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this whole distance over here, which is the glass portion

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plus the wood trim on either end, it adds 8 inches to the

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glass portion.

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The glass portion is 3x, so we're going to add 8 to it.

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So this height right here 3x plus 8.

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And now let's think about the width.

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The width of the glass portion can be represented as 2x, so

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that distance right there, just the glass portion, is 2x.

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But then they say when you add the wood trim, the wood trim

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adds 7 inches to the total width, so this total width

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right here, because we're adding that piece and that

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piece to it, is 2x plus 7.

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That is the total width of the window, the total

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height is 3x plus 8.

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Now, what they want us to do is write a polynomial

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expression that represents the total area of the window

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including the glass and wood.

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Simplify the expression.

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Now, the total area is just going to be the

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height times the width.

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So let's just write that down.

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The area is just going to be equal to the height, which is

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3x plus 8 times the width, times 2x plus 7.

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And that is the expression we've written up.

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Well, we haven't done it as a polynomial.

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We've written it as a product of two binomials.

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And if we want to simplify it, we can just multiply it out.

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So the way you can think about it is we multiply-- you're

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doing the distributive property twice essentially.

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But you could say let's multiply 3x times 2x, so you

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get 3x times 2x.

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That's multiplying it times that.

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Then you want to multiply 3x times 7, which is 21x-- let me

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just write it this way-- plus 3x times 7.

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And then you want to multiply this 8 times both of these

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terms. So then you have plus 8 times 2x.

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Let me do this.

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So plus 8 times 2x and then you have plus 8 times 7.

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And now we just have to simplify everything.

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So what does this become?

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3x times 2x is 6x squared plus 3x times 7 is plus 21x plus 8

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times 2x, so plus 16x, and then plus 8 times 7, 8 times 7

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is 56, plus 56.

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We can combine these two middle terms.

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21x plus 16x is what?

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37x.

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So then we get-- let me just do it in another color-- 6x

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squared plus 37x plus 56.

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Now we've expressed it as a simplified polynomial

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expression.

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