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  • - [Instructor] We are told a rectangle has a height of five

  • and a width of 3x squared minus x plus two.

  • Then we're told express the area

  • of the entire rectangle,

  • and the expression should be expanded.

  • So pause this video and see if you could work through this.

  • All right, now let's work through this together.

  • What this diagram is showing us

  • is exactly an indicative rectangle,

  • where its height is five

  • and its width is 3x squared minus x plus two.

  • What this shows us is is the area

  • of the entire rectangle can be broken down

  • into three smaller areas.

  • You have this blue area right over here.

  • Where the width is 3x squared, the height is five,

  • so what's that area going to be?

  • It's going to be the height times width.

  • Five times 3x squared.

  • That's the same thing as 15x squared.

  • Now what about this purple area right over here?

  • It's going to be height times width again,

  • so it'll be five times negative x,

  • which is the same thing as negative 5x.

  • I know what some of you are thinking,

  • how can you have a width of negative x?

  • Well, we don't know what x is,

  • so this is all a little bit abstract.

  • But you could imagine having x be a negative value,

  • in which case this actually would be a positive width.

  • Another thing you might be saying is,

  • hey, this magenta area, when I just eyeball it,

  • looks like it has a larger width than this blue area.

  • How do we know that?

  • We don't.

  • They're just showing this as an indicative way.

  • They might have actually the same width.

  • They might have, one might be larger than the other,

  • but this is just showing us that

  • they're not necessarily the same.

  • It's very abstract.

  • It's definitely not drawn to scale

  • 'cause we don't know what x is.

  • All right, and then this last area

  • is going to be the height, which is five,

  • times the width, which is two.

  • So that's just going to be equal to 10.

  • So what we just saw is that the area of the whole thing

  • is equal to the sum of these areas,

  • and the sum of those areas is 15x squared

  • plus negative 5x, or we could just write that as minus 5x.

  • And then we have plus 10.

  • Now I know what some of you are thinking.

  • If I know that the height is five

  • and the width is this value,

  • well couldn't I have just multiplied height

  • times the entire width, 3x squared minus x plus two,

  • and then I would've just naturally distributed the five?

  • And essentially that's exactly what we did here.

  • Area models, you might have first

  • seen them in elementary school,

  • really to understand the distributive property.

  • So if you distribute the five you get 15x squared

  • minus 5x plus 10.

  • Same idea.

  • Let's do another example.

  • So here we're given another rectangle.

  • It has a height of 2x.

  • We see that right there.

  • A width of 3x plus four.

  • We see that over there.

  • Express the area of the entire rectangle.

  • So same drill.

  • See if you can pause this video

  • and work on this on your own.

  • All right.

  • Well, we can see that the height is 2x,

  • the width in total is 3x plus four.

  • But we can clearly see that the area of the entire thing

  • can be broken up into this blue section

  • and this magenta section.

  • The blue section's area is 3x times 2x.

  • So 3x times 2x, which is equal to 6x squared.

  • The magenta section is equal

  • to width times height, its area.

  • So that going to be equal to 8x.

  • So the area of the whole thing is going

  • to be the sum of these areas,

  • which is going to be 6x squared plus 8x.

  • And we're done.

  • Once again, we could've just thought about it as

  • the height is 2x, we're gonna multiply it times the width,

  • times 3x plus four.

  • Then we just distribute the 2x.

  • So 2x times 3x is 6x squared.

  • 2x times four is 8x.

  • Same idea.

  • It's really just to make sure we understand

  • that when we distribute a 2x onto this 3x and this four,

  • we can conceptualize it as figuring out

  • the areas of the sub-parts of the entire rectangle.

- [Instructor] We are told a rectangle has a height of five

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