Name: Area model for multiplying polynomials with negative terms
Uploaded: 2020-03-27T17:39:56.000Z
Duration: 5 min 17 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Gerund

we've already looked at using area models

To-infinitive

like multiplying x plus seven times x plus three.

In those videos, we saw that we could think about it

where we could break up the length of the rectangle

and then the rest of it has length seven.

So this would be seven here, and then the total length

And then the total length of this side would be x plus,

And what area models did is they helped us visualize

Because if we're looking for the entire area,

the entire area is going to be x plus seven,

And then of course, we can break that down

This rectangle, and this is actually going to be a square,

This one over here will have an area of seven x,

This one over here will have an area of three x.

And then this one over here will have an area

And so we can figure out that the ultimate product here

That's going to be the area of the entire rectangle.

What if we were dealing with negatives instead?

For example, if we now try to do the same thing,

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.

So we could write an x there for that part of the height.

So let's see, if we kept going with what we did last time,

The area here would be negative seven times 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

add these two together to get negative 10x.

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,

and so we would take it away from the whole.

And that's exactly what is happening in this expression.

that even before when this wasn't a negative seven,

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,

I'll give an example, if x were equal to 10.

we would get an area model that looks like this.

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 we're gonna take it away from the total area.

This green area is negative three times 10,