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We're told to factor 4x to the fourth y, minus 8x to the

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third y, minus 2x squared.

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So to factor this, we need to figure out what the greatest

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common factor of each of these terms are.

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So let me rewrite it.

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So we have 4x to the fourth y, and we have minus 8x to the

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third y, and then we have minus 2x squared.

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So in the other videos, we looked at it in terms of

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breaking it down to its simplest parts, but I think we

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have enough practice now to be able to do a little bit more

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of it in our heads.

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So what is the largest number that

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divides into all of these?

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When I say number I'm talking about the actual, I guess,

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coefficients.

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We have a 4, an 8, a 2.

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We don't have to worry about the negative signs just yet.

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And we say, well, the largest, of, the largest common factor

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of 2, 8 and 4 is 2.

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2 goes into all of them, and obviously that's the largest

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number that can go into 2.

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So that is the largest number that's going to be part of the

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greatest common factor.

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So let's write that down.

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So it's going to be 2.

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And then what's the greatest, I guess, factor, what's the

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greatest degree of x that's divisible into

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all three of these?

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Well, x squared goes into all three of these, and obviously

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that's the greatest degree of x that can be divided into

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this last term.

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So x squared is going to be the greatest common x degree

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in all of them.

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2x squared.

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And then what's the largest degree of y that's divisible

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into all of them?

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Well, these two guys are divisible by y, but this guy

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isn't, so there is no degree of y that's

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divisible into all of them.

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So the greatest common factor of all three of these guys

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right here is 2x squared.

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So what we can do now is we can think about each of these

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terms as the product of the 2x squared and something else.

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And to figure that something else we can literally

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undistribute the 2x squared, say this is the same thing, or

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even before we undistribute the 2x squared, we could say

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look, 4x to the fourth y is the same thing as 2x squared,

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times 4x to the fourth y, over 2x squared.

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Right?

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If you just multiply this out, you get 4xy.

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Similarly, you could say that 8x to the third y-- I'll put

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the negative out front-- is the same thing as 2x squared,

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our greatest common factor, times 8x to the third y, over

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2x squared.

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And then finally, 2x squared is the same thing as if we

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factor out 2x squared-- so we have that negative sign out

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front-- if we factor out 2x squared, it's the same thing

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as 2x squared, times 2x squared, over 2x squared.

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This is almost silly what I'm doing here, but I'm just

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showing you that I'm just multiplying and dividing both

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of these terms by 2x squared.

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Multiplying and dividing.

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Here it's trivially simple.

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This just simplifies to 2x squared right there, or this

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2x squared times 1.

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That simplifies to 1, maybe I should write it below.

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But what do these simplify to?

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So this first term over here, this simplifies to 2x squared

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times-- now you get 4 divided by 2 is 2, x to the fourth

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divided by x squared is x squared.

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And then y divided by 1 is just going to be a y.

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So it's 2x squared times 2x squared y, and then you have

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minus 2x squared times, 8 divided by 2 is 4.

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x to the third divided by x squared is x.

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And y divided by 1, you can imagine, is just y.

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And then finally, of course, you have minus 2x squared

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time-- this right here simplifies to 1-- times 1.

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Now, if you were to undistribute 2x squared out of

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the expression, you'd essentially get 2x squared

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times this term, minus this term, minus this term.

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Right, if you distribute this out, if you take that out of

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each of the terms, you're going to get 2x squared times

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this 2x squared y, minus 4xy, and then you have minus 1,

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minus 1, and we're done.

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We've factored the problem.

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Now, it looks like we did a lot of steps.

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And the reason why I kind of of went through great pains to

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show you exactly what we're doing is so you know exactly

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what we're doing.

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In the future, you might be able to do this

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a little bit quicker.

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You might be able to do many of the steps in your head.

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You might say OK, let me look at each of these.

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Well, the biggest coefficient that divides all of these is a

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2, so let me put that 2, let me factor 2 out.

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Well, all of these are divisible by x squared.

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That's the largest degree of x.

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Let me factor an x squared out.

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And this guy doesn't have a y, so I can't factor a y out.

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So let's see, it's going to be 2x squared times-- and what's

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this guy divided by 2x squared?

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Well 4 divided by 2 is 2.

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x to the fourth divided by x squared is x squared.

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y divided by 1, there's no other y degree that we

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factored out, so it's just going to be a y.

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And then you have minus 8 divided by 2 is 4.

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x to the third divided by x squared is x.

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And then you have y divided by say, 1, is just y.

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And then you have minus 2 divided by 2 is 1.

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x squared divided by x squared 1, so 2x squared divided by 2x

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squared is just 1.

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So in the future, you'll do it more like this, where you kind

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of just factor it out in your head, but I really want you to

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understand what we did here.

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There is no magic.

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And to realize that there's no magic, you could just use the

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distributive property to multiply this out again, to

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multiply it out again, and you're going to see that you

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get exactly this.

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