 This video will talk about the greatest common factor. When you're looking for the greatest common factor, you're actually looking for the largest factor to all the terms that you're trying to factor. So if we start out with just numbers, if we want to find the greatest common factor, the best thing to do would be just to start listing all the factors of 18. So 1 times 18 and 2 times 9 and 3 times 6 and then we have 6 times 3 but we've already listed that combination. So there's all the factors of 18 and 24 would be 1 times 24 and 2 times 12 and 3 times 8 and 4 times 6 and then it would be 6 times 4. Now we're working our way back the other direction. So we're looking to compare what they have greatest in common. They both have 1 and they both have 2 and they both have 3 and they both have 6 and after 6 there's really nothing else in common. 24 doesn't have 18 or 9 so the greatest common factor here is equal to 6. Let's try again. 45 would be 1 times 45. Now you might be able to do some of these just in your head. That's a 45 and 2 doesn't go in because it's not even but 3 does 3 times 15 is 45. 4 doesn't 5 does 5 times 9 and then 9 would be the next factor. 27 would be 1 times 27 and 2 doesn't but 3 does 3 times 9 and then it's going to be 9 times 3. So this only has 4 factors. 45 has 6 different factors and we look for the greatest common one. It looks like the greatest one they have in common would be 9. They all have 1, 3 and 9 but then there's just 27 left here which is not a factor of 45 so the greatest common factor is going to be 9. Now what happens when we have variables? With variables you want to take the smallest exponent because it has to be common to both of them. So I have an A and I have 2 factors of B on this for AB and for A cubed, B cubed I have A times A times A and B times B times B. And when I try to find the greatest common factor the only amount of A's that they have in common is 1 and they have 2 B's in common. So the greatest common factor is going to be A and then the B squared, 2 factors of B. And notice A with an exponent of 1 is the smallest between 1 and 3 and 2 on the B is the smallest exponent between 2 and 3 so it's the smallest exponent again. Now in this example we have numbers and letters. Let's take care of the letters first because it's a little bit easier. We've got a 2 and we've got a 1 and we've got a 4. So X is going to be the smallest exponent and for the Y's we've got a 3 as an exponent and a 2 as an exponent in both cases in these last two cases. So we're going to have our greatest common factor down here is going to have some number and then X, Y squared. Now we have to consider 8, 12 and 16. So again we do 1 times 8, 2 times 4. That's all there are for factors of 8. 12 would be 1 times 12 and 2 times 6 and 3 times 4 and then 16 is going to be 1 times 16, 2 times 8 and 4 times 4. And it looks like 4 is going to be our greatest common factor that's common to all 3 and the biggest thing that's common to all 3. So our greatest common factor then would be equal to 4 X, Y squared. Let's see if we can put this to use. Kind of get the idea of how to use a greatest common factor to factor. So 5X plus 15. I'm going to say that I have X plus 3 so what do I need? I have just an X and I want to get to 5X then I need a 5 out here. So 5 times X would give me 5X. And 1 times 3 would give me 15. Well that would be 5 times 3. So it must be that we had a common factor of 5. Now we did, I did part of it here for you but let's do it this way now. 3, 9 and 6. They're going to have 3 in common. And then A cubed A and A squared we went the smallest exponent so we just went A. So now we ask ourselves we have a factor of 3 and an A. And I have 3A but I need 2 more factors of A so I have to put those inside. And if I were to distribute here I'd have 3A cubed. So 3 times what will give you negative 9? Well that would be negative 3. Negative 3 times 3 will give you negative 9 and I have A on the outside, 1 factor of A so I don't need any more factors of A on the inside. And then finally we have that plus 6A squared. So 3 times positive 2 would give me the positive 6A. And then I have A, one factor on the outside but I need a total of 2 factors so the other factor has to go inside. So 3A times the quantity A squared minus 3 plus 2A. I don't know what happens if we have 2 terms but they've got more involved looking things. We've got this X times X minus 7 that's one term. And I've got 2 times X minus 7 that's my second term. So I'm looking to see what they have in common. And what it looks like this is remember X times this quantity X minus 7 and that's 2 times this quantity X minus 7. So the X minus 7 quantity is what they both have in common. It's common to both of them. So greatest common factor is X minus 7 and if I want to factor it I would say I have that X minus 7 and then we ask ourselves again. Okay well I have X minus 7 but I need another factor of X so that needs to go inside my second factor. And I have X minus 7 but what times X minus 7 will give me 2 times X minus 7? Well it's a positive 2 so that would be my other factor. So X minus 7 is the common factor and then I have leftovers. X plus 2. Let's try again. There's 2 terms here. So I have 5 times X minus 1 and I've got negative X times X minus 1. So in this case if I just look at this comparing it to that one they have X minus 1 in common. That's my greatest common factor. So X minus 1 is a factor and then X minus 1 times 5 will give me this first term so I need a 5 inside that other parenthesis and then I'm going to look at this term here and X minus 1 times negative X is what gives me that second factor so I need a minus X in my other factor.