Main content

### Course: Algebra 1 > Unit 13

Lesson 1: Multiplying monomials by polynomials# Multiply monomials by polynomials: Area model

We can use an area model to multiply a polynomial by a single term. Created by Sal Khan.

## Want to join the conversation?

- At3:19Sal says that 2x times 3x is equal to 6x^2. The reason why he put the ^2 is because he is solving for the area right? if thats true then how come 2x times 4 isnt equal to 8x^2? Please let me know if im wrong(13 votes)
- Think about it: 2x * 3x can be rewritten as 2 * 3 * x * x.

This becomes 6x^2. 2x * 4 can be rewritten as 2 * 4 * x. This becomes 8x. Hope this helps!(40 votes)

- yo yall trying to play some 2k21 rq(10 votes)
- yeah bro whats your username(12 votes)

- dis leson makky myy bwain hurty(8 votes)
- the area in the first example is -9.16666666666+3.9965I

it's a negative number !?(4 votes)- Where did you get that number? Also, the rules of polynomial is that the exponents should not be negative.(1 vote)

- At0:53, are we basically adding up all of the areas? Like what exactly is going on T_T(3 votes)
- You solve for each individual section and put them in an equation, but don't solve. Comment if this doesn't make sense(3 votes)

- I cannot figure it out. What is the sum of 6x2+8x? what is than the outcome? thank in advance(3 votes)
- You'll see a lot in algebra that often times you won't see any sum. This is one of such scenarios(3 votes)

- What is the difference between a trinomial and a polynomial?(2 votes)
- A trinomial is a term used specifically for a polynomial with 3 terms. The difference is that a Trinomial is a specific kind of Polynomial, while a Polynomial is the general description of 'nomials' with a definite number of terms(3 votes)

- Where do you get the 2 from in the second question(3 votes)
- How is everyone doing today :D(2 votes)
- Been better, been worse... kinda going through pre-teen phase, yk?(2 votes)

## Video transcript

- [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.