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We're asked to factor 20u squared v minus 10uv squared.

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And when they say factor a binomial like this, an

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expression that has two terms like this, they really mean

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break it up into the product of one or more terms. So let's

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see if we can do that.

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And the easiest way we can do that, is say hey, is there any

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common factor?

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In particular, let's find the greatest common factor of each

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of these terms and then divide that out.

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Or you can almost imagine, un-distribute it out, and I'll

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show you what I'm talking about in a second.

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And eventually you'll be able to do this in your head, but

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we'll work through it step by step right now.

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So what is 20u squared v if we factor it out?

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20u squared v, if we do into the prime factorization, it is

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20 is 2 times 2 times 5.

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Right?

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That's 2 times 10, that's 20.

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u squared is times u times u and then v is just one v.

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So we just rewrote 20u squared v into kind of a product of

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its smallest components, its most fundamental components,

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prime numbers and just u's and v's.

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Now let's do the same exact thing for the 10uv squared.

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So we'll put that minus sign there so that we haven't

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fundamentally changed the expression.

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10 is, if we break it down to its prime factors, 2 times 5.

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Then we have one u times one u times v times v.

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That's what v squared is, times v times v.

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Now what's the greatest common factor of these two terms

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right here?

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Well let's see.

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They both have one 2.

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They both have one 2 right there.

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Maybe I'll circle that one, I could have circled that one.

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They both have one 5-- that's one 5, one 5.

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They both have one u, one u there, one u there.

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This one has two, but only this one has one, and they

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both have at least one v.

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So the greatest common factor is 2 times 5 times u times v.

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So I could rewrite this expression, I can kind of

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un-distribute the 2 times 5 times u times v,

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and what'll we get?

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If we wrote 2 times 5 times u times v, and we say that's

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going to be-- this expression is equal to this times what?

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Well if you factor the 2 times 5 times u times v out, all

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you're going to be left with in this first term is the 2

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times u, so 2u here.

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And in the second term, all you're going to be

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left with is a v.

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Right?

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All this other stuff gets factored out.

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All you're going to be left with is a v.

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Hopefully you see, if I multiply 2 times 5 times u

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times v times 2u, I'm going to get this first term here.

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So if I were to distribute it, I would get this first term.

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And if I multiply 2 times 5 times u times v times this v

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over here, I'm going to get this second term.

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So this expression, and that expression is

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the exact same thing.

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We have factored it out, now we can

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simplify it a little bit.

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2 times 5 times u times v we rewrite as 10uv.

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And then inside the parentheses, we of course have

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a 2u and then a minus v.

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And we're done!

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We have factored the expression.

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Now you won't be doing it to this granular level, but this

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is the best way to think about it.

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Eventually you're going to say, hey, wait, look, the

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largest number that divides both of these is a 10.

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Because you could see 10 goes into the 20, 10 goes into 10.

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And, let's see, a u goes into both of these, and a v goes

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into both of these.

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So let me factor out a 10uv, and then if I divide this

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thing by 10uv, I'm going to be left with 2u.

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And if I divide this by 10uv, I'm going to just

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be left with a v.

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So that's another way to think about it.

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Let me do that right now, so we could say that this is the

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same thing.

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Another way of approaching it, you could have said that this

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is the same thing as-- Well, the largest number that

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divides both of these is 10uv, and that's going to be times

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20u squared v over 10uv minus this thing.

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10uv squared over 10uv.

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This expression and this are obviously the same thing.

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If I were to distribute the 10uv it would cancel out with

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each of these in the denominator right there.

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So they're the same thing, but we can do is we

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can simplify this.

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We could say that 20 divided by 10 is just 2, u squared

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divided by u is just a u, v divided by v is just 1, 10

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divided by 10 is 1, u divide by u is u, v squared divided

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by v is just a v to the first power.

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So you're left with 10uv times the quantity 2u minus v.

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Either way you get the same answer.

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