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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.

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  • blobby green style avatar for user Akira
    The last example at around :

    (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.
    (9 votes)
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    • starky sapling style avatar for user 𝘽𝘼𝙏𝙈𝘼𝙉
      Let's forget this is an area model for a minute, you see the equation above the area model at ? 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
      (21 votes)
  • purple pi purple style avatar for user Shaurya K
    What happens if x is between 4 and 6? The area would be negative... So, how does that work out?
    (4 votes)
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    • blobby green style avatar for user ShellyKBernard
      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.
      (2 votes)
  • piceratops ultimate style avatar for user Philosopher King
    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?
    (4 votes)
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    • duskpin ultimate style avatar for user Hat Girl
      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)
      (3 votes)
  • aqualine seed style avatar for user Andrea
    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)
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  • blobby green style avatar for user Atarah Smith
    i don't undersand why the answer for both equations is 21
    (4 votes)
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  • stelly yellow style avatar for user inmydreamsin
    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)
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    • blobby green style avatar for user Aldana
      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.
      (1 vote)
  • piceratops seedling style avatar for user Drew
    how would I use a poly/trinomial in real life?
    (3 votes)
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  • leaf blue style avatar for user Smit
    How can you have a negative area?
    (2 votes)
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  • stelly blue style avatar for user naomi davis
    how do you get 21
    (1 vote)
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  • blobby green style avatar for user Miracle Lopez
    I dunno how to add up all of the equation numbers that I got. like I have the correct numbers but I dunno how to get the full total out of all of them. I keep getting it wrong.
    (1 vote)
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    • piceratops sapling style avatar for user Eliza
      If you're still having trouble with this, remember to only combine like terms which have the same variable and exponent. If you post a polynomial you're having trouble with and show the steps you took to complete it, it would help to understand where you're going wrong.
      (1 vote)

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.