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### Course: Algebra 2>Unit 1

Lesson 4: Multiplying monomials by polynomials

# 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 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.
• 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!

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
• What happens if x is between 4 and 6? The area would be negative... So, how does that work out?
• 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.
• 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?
• 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)
• 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.
• Hmm, weird, It works fine for me... Reload your internet maybe?
• How can you have a negative area?
• 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.
• 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
• 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.
• i don't undersand why the answer for both equations is 21
• 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?
• 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?
• 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!