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Find the greatest common factor of these monomials.

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And when they say monomials, that's just a fancy word for

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saying a one term expression.

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Each of these only, obviously, have one term in them.

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Now to find the greatest common factor of these, the

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way I think about it is, I like to break up each of these

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terms into their constituent parts.

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Make them a product of the simplest things possible.

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For regular numbers like 10, to me that means break them up

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into their prime factors, and for these variable expressions

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like cd squared, break it up into the product of the most

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simple variable, for example, c times d times d.

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So let's do that for each of them and see what the greatest

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common factor is.

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Where do these overlap in terms of their factor?

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And we care about the greatest overlap.

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So let's do this first one.

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10 cd squared, what is that equal to?

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Well 10 is equal to 2 times 5.

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You could do a factoring tree here, but these are pretty

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straightforward numbers to factor into.

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They're prime factors.

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So 10 is 2 times 5, c, all you can do is break that, you

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could just write that as a c, you can't really

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simplify that anymore.

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And d squared can be written as d times d.

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So I have essentially broken 10cd squared into this, into

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the product of kind of the smallest constituents that I

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could think of.

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The prime factors of 10, and then c, and then d.

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Now let's do 5cd.

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Well 5cd, 5 is prime, so its prime factorization is

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literally just 5.

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c you can't break that down anymore, that's just a c, and

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then times a d.

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So we really didn't do anything to this expression

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right there.

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And then finally you have 25c to the third d squared.

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Well 25 is 5 times 5, and then we have times c times c times

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c, that's what c to the third is, and then we have times d

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squared, times d times d.

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Now, what is the greatest common factor, or what is the

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greatest common overlap between these three things?

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Well they all have a 5.

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Let me circle them.

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You have a 5 there, you have a 5 there, you have a 5 there.

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They all have at least one c.

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You have one c there, one c there, and

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then another c there.

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And they all have at least one d.

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You have a d there, you have a d there, and then

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you have a d there.

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Now they don't all have a second d, only the first one

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and the third one have a second d.

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And they all don't have a second or third c, only this

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last one has a second or third c.

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So we're essentially done.

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The greatest common factor is 5cd.

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In fact you can't have a greater number than 5cd be a

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common factor, because the largest factor of 5cd is 5cd.

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So the greatest common factor of these three monomials, or

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these three expressions, is 5cd.

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The largest number of factors that overlaps with all three

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of these expressions is a 5, one c, and one d.

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