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### Course: Algebra 2 > Unit 1

Lesson 4: Multiplying monomials by polynomials- Multiplying monomials
- Multiply monomials
- Multiplying monomials by polynomials: area model
- Area model for multiplying polynomials with negative terms
- Multiply monomials by polynomials: area model
- Multiplying monomials by polynomials
- Multiply monomials by polynomials
- Multiplying monomials by polynomials review

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# Area model for multiplying polynomials with negative terms

Discover how to multiply monomials by polynomials using area models. This method works even when dealing with negative terms! By visualizing the process, we can understand why we multiply different terms and how negative areas affect the total area.

## Want to join the conversation?

- The last example at around5:00:

(10 - 7) * (10 - 3)

= 100 -70 -30 +21

My question:

=> How do you explain the`+21`

in the last equation?

I know algebraically`negative * negative = positive`

.

However,`negative length * negative length = positive area`

is not so intuitive in this example to me...

What I can understand is:

=>`100`

is the area of`10 * 10`

square.

=>`-70`

and`-30`

is the area that's substracted from the entire area.(10 votes)- Let's forget this is an area model for a minute, you see the equation above the area model at5:00? Sals just showing you how to input the terms in an area model!

and it works! an area is*a = Width*Height*

so... the*Width*of the top cube is x! and the*Height*is x!

multiply! x squared! right?

same with the cube under it!*Height*= -3*Width*= x ...... -3x!

and the pink cube!*Height*= -7*Width*= x ...... -7x!

you asked about how we got 21, this is how!

*Height*= -3*Width*= -7... multiply!

the solution is 21!

but wait... how did we get 10? sal is just giving you an example if x was equal to 10!

hope this helped! :D(29 votes)

- What happens if x is between 4 and 6? The area would be negative... So, how does that work out?(4 votes)
- Remember, what we are looking at is a quadratic function, which is a parabola on the graph. The roots on the graph can be pulled from the factors, +3 and +7. Because the leading coefficient is positive, +1, we know the parabola is convex so "area" or the y coordinate, will be negative between those roots. Graph the function in desmos to visualize this "area" function. Negative area doesn't necessarily mean anything tangible. It means there is a void of a certain amount of space. How much void is determined by your input, the x value, between the roots where that area becomes a negative value.(3 votes)

- In the last example, what is 21 supposed to be? It seems like Sal is saying that 21 is the area of the entire rectangle, but how can the area of the entire rectangle be 21 when it's obvious that only that bottom right corner is 21? How can something that is clearly bigger than that small area be equal to it? Or is 21 not the area of the entire rectangle?(5 votes)
- The negative area part is kind of misleading but yes, the answer is 21. Since you can't have a negative side length or area (as far as I know,) the negative side lengths are subtracted from x. Since x is 10, the side lengths become 7 and 3.

(sorry if I sound kinda condescending)(4 votes)

- There's a grey screen. I have already reloaded twice and the video will be there but it will only play for one second and turn into a grey screen again.(4 votes)
- Hmm, weird, It works fine for me... Reload your internet maybe?(5 votes)

- How can you have a negative area?(3 votes)
- Sal presents the discussion as a negative area being taken away from the rectangle, so you would have the large rectangle as the outline shape, but the upper right and lower left would be "taken away" so they would be in dashed lines rather than solid.(5 votes)

- Why in the picture, we subtract 70 and 30 from
**rectangle**?

It's total aria of (10+7)x(10+3) which equal 221.

We need subtract 70 and 30 from square 10x10. Isn't it?

Initial shape must be**square**. 10x10=100 Then we subtract two arias 10*7 and 10*3. Then because we subtract (3*7)=21 twice - fist with 10*7, second with 10*3, we need compensate for this + 21

100-70-30+21=21(4 votes)- Well, in the first example, our first equation was x^2 - 7x - 3x + 21. All Sal did was replace x with 10 as an example, which gives us the equation: 100 - 70 - 30 + 21. Now, we would solve this equation from left to right. 100 - 70 = 30, and 30 - 30 would give us 0. The only thing that is left is the 21, so 0 + 21 = 21. Therefore, making 21 our final answer.(2 votes)

- i don't undersand why the answer for both equations is 21(4 votes)
- If you are referring to (x+7)(x+3) and (x-7)(x-3) then forget the x for a bit. 7 times 3 equals 21. And in the second one: -7(-3)= 21.(1 vote)

- how would I use a poly/trinomial in real life?(3 votes)
- Why did Sal use an Area Model for using negative numbers wouldn't it have been easier to understand if it was like a graph somehow?

Also, how can you have negative area's as seen in the second example?(2 votes)- Hey Danny,

Using an area model for multiplying polynomials, including those involving negative numbers, can be a helpful visual tool because it allows you to break down the multiplication process into simpler steps. The area model essentially represents each term of the polynomial as a rectangle, and then you find the area of each rectangle and add them up to get the total area, which corresponds to the product of the polynomials. This method provides a tangible way to understand the distributive property and the interactions between the terms.

Regarding your question about negative areas, it might seem counterintuitive at first. Negative areas arise when you have a negative value for one of the dimensions in the area model. In terms of the polynomial multiplication example, a negative coefficient essentially reverses the orientation of the rectangle representing that term. So, when you calculate the area of that rectangle, the result is negative. This concept can be a bit abstract, but it aligns with the idea that negative numbers can represent quantities below zero or in the opposite direction.

Hope this helps!(2 votes)

- how do you get 21(1 vote)
- A minus times a minus is a positive(3 votes)

## Video transcript

- [Instructor] In previous videos, we've already looked at using area models to think about multiplying expressions, like multiplying x plus
seven times x plus three. In those videos, we saw
that we could think about it as finding the area of a rectangle, where we could break up
the length of the rectangle as part of the length has length x, and then the rest of it has length seven. So this would be seven here,
and then the total length of this side would be x plus seven. And then the total length of
this side would be x plus, and then you have three right over here. And what area models did
is they helped us visualize why we multiply the different terms or how we multiply the different terms. Because if we're looking
for the entire area, the entire area is going
to be x plus seven, x plus seven times x plus three, times x plus three. And then of course, we can break that down into these sub-rectangles. This rectangle, and this is
actually going to be a square, would have an area of x squared. This one over here will
have an area of seven x, seven times x. This one over here will
have an area of three x. And then this one over
here will have an area of three times seven, or 21. And so we can figure out that
the ultimate product here is going to be x squared plus seven x plus three x plus 21. That's going to be the area
of the entire rectangle. Of course, we could add the seven x to the three x to get to 10x. But some of you might be wondering, well, this is all nice when I have plus seven and plus three. I can think about positive lengths. I can think about positive areas. But what if it wasn't that way? What if we were dealing
with negatives instead? For example, if we now
try to do the same thing, we could say, all right, this top length right over here would be x minus seven. So let's just keep going with it, and let's call this length
negative seven up here. So it has a negative seven
length, and we're not necessarily used to thinking
about lengths as negative. Let's just go with it. And then the height right over here, it would be x minus three. So we could write an x there
for that part of the height. And for this part of the height, we could put a negative three. So let's see, if we kept going
with what we did last time, the area here would be x squared. The area here would be
negative seven times x, so that would be negative seven x. This green area would be negative three x. And then this orange area
would be negative three times negative seven,
which is positive 21. And then we would say that the
entire product is x squared minus seven x minus three x plus 21. And we can, of course, add these two together
to get negative 10x. But does this make sense? Well, one way to think about
it is that a negative area is an area that you would
take away from the total area. So if x happens to be
a positive number here, then this pink area would be negative, and so we would take
it away from the whole. And that's exactly what is
happening in this expression. And it's worth mentioning that even before when this
wasn't a negative seven, when it was a positive seven and this was a positive seven x, it's completely possible
that x is negative, in which case you would've
had a negative area anyway. But to appreciate that
this will all work out, even with negative numbers, I'll give an example,
if x were equal to 10. That will help us make sense of things. So if x were equal to 10, we would get an area model
that looks like this. We're having 10 minus seven, so I'll put minus seven right over here, times 10 minus three. Now, you can figure out in your heads what's that going to be. 10 minus seven is three. 10 minus three is seven. So this should all add up to positive 21. Let's make sure that's actually occurring. So this blue area is going to
be 10 times 10, which is 100. This pink area now is
10 times negative seven. So it's negative 70, so we're gonna take it
away from the total area. This green area is
negative three times 10, so that's negative 30. And then negative three
times negative seven, this orange area is positive 21. Does that all work out? Let's see, if we take this
positive area, 100 minus 70 minus 30 and then add 21, 100 minus 70 is going to be 30, minus 30 again is zero, and then you just have 21 left over, which is exactly what you would expect. You could actually move
this pink area over and subtract it from this blue area. And then you could take this green area and then you could turn it vertical, and then that would subtract
out the rest of the blue area. And then all you would have
left is this orange area. So hopefully this helps you appreciate that area models for multiplying
expressions also works if the terms are negative. And also, reminder, when
we just had x's here, they could've been negative to begin with.