Multiply monomials by polynomials: Area model | Algebra 1 | Khan Academy

We are told a rectangle has a height of five and a width of three x squared minus x plus two. Then we're told to express the area of the entire rectangle, and the expression should be expanded. So pause this video and see if you can 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 5 and its width is three x squared minus x plus two. What this shows us is that 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 and the height is 5. So what's that area going to be? Well, it's going to be the height times the width: 5 times 3x squared. And that's the same thing as 15x squared.

Now, what about this purple area right over here? Well, it's going to be height times width again, so it'll be 5 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; 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 because we don't know what x is.

All right, and then this last area is going to be the height, which is 5, times the width, which is 2. 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 15 x squared plus negative 5x. 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 5 and the width is this value, well, couldn't I have just multiplied height times the entire width, 3x squared minus x plus 2? Then I would have just naturally distributed the 5. 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. If you distribute the 5, you get 15 x squared minus 5x plus 10. Same idea! Let's do another example.

So here we're given another rectangle that has a height of 2x. We see that right there, and a width of 3x plus 4. 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 and the width in total is 3x plus 4. 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 is 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 have just thought about it as the height is 2x. We're going to multiply it times the width, 3x plus 4, and then we just distribute the 2x.

So 2x times 3x is 6x squared, and 2x times 4 is 8x. Same idea! It's really just to make sure we understand that when we distribute a 2x onto this 3x and this 4, we can conceptualize it as figuring out the areas of the sub-parts of the entire rectangle.