Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Common factors. Sometimes we want to take polynomials and make them into monomials. A monomial being something that's only multiplied and or divided. So in this example, we're going to look and see if there's anything that they all have in common that we could factor out. We can see that two -4 and +6 would all have a factor of 2, and then X to the 4th, X ^3 and X would all have an X. If I pull a 2X out, what I want to think about is 2X times what gives me two X to the 4th, and the answer would be X ^3, 2 X times what gives me -4 X cubed. The answer would be -2 X squared and two X times. What would give me 6X. Now sometimes I have seen people do things like write it this way. If I'm going to factor out a 2X then I'm really thinking about dividing each term by two X and if I divide each term by two X then I can actually simplify. So two into two X into X to the 4th would leave me X ^3. 2 into 4 gives me a 2X into X ^3, X ^2 and we can see that we did get the same answer. So sometimes people like to write it as a division and simplify each term. This way if we pull out the 2X, it's really the same thing as if I'm pulling out a 2X, I'm dividing each of them by two X. This next example, what does 3X and six have in common? Well, they each have a three. If I pull out a three, three times what gives me 3X, and three times what gives me six? Once again, you could think of it as dividing each term by that three once you pull it out if you want, and then simplifying from there. You'll get the same answer either way. This next one, Well, what if we have an X plus A5? YX and five Y really don't have anything in common except for an understood one. So every single term can be turned into a monomial, every polynomial can be a monomial, and that happens just by factoring out of one. Sometimes you see things that say can't factor, and mathematically that's actually not accurate. Everything can factor. I can always pull out of one because one times anything is itself. This next example gets a little different because we actually are starting with two terms. So this whole thing that's multiplied together with our distributor property is one term and then there's a plus that separates our terms, pluses and minus separate terms and then a second term. So when I look at these two terms, what do they have in common? And hopefully we see that they both have AB plus C If I take out AB plus C from each term, what do we get left? We get left A + D. Now this is where that dividing each term might make a little easier to see. If I have this A * B + C and I divide it by b + C and I have AD times b + C and I divide by B + C, we know anything over itself, IE this B + C / B + C cancels and is 1. So that would leave A as that first term in the parenthesis, and then this B + C and this B + C are going to cancel, leaving the D This next one, we start with four terms. So this last one, if I rewrite it, we get B + C times the quantity A + D This next one, we have 4 terms, one term plus or minus another term, plus or minus another, plus or minus another. We're going to do grouping in this case, we're going to start by looking to see, first of all, do all four of them have anything in common? And the answer is no. So we're going to group them two and two. So if I look at these first two terms, do they have anything in common? And the answer is yes, they have AB. So if I factor out of BI get A + C left, if I look at the second two terms, do they have anything in common? And the answer is yes, they have an E. And that leaves me an A + C So we went from 4 terms and now we're at two terms. So we're not done. We need to make this into one term or a monomial. So this term here and this term here. Do these two terms have anything in common? And the answer is yes, they each have an A + C If I pull out an A + C from each of the terms, that leaves me AB plus E in my second set of parentheses. That is now one term, and hence it's factored all the way. This last example, there are four terms. All four terms don't have anything in common because if they did, we'd start by pulling that out first. So we're going to group the 1st 2 together and have an A ^2, and that's going to leave me an A -, 3. Now, these second two terms, I want to have an A -, 3 in my parentheses or what's left. This one's a little tricky. We have to think it through. We're actually going to pull out a -2 because if I pull out a -2, that changes my signs for the -2 A+ 6, and now it turns into a -, 3. So I went from 4 terms in the original to two terms. Now. These two terms each have an A -, 3 in common. So if I write down what's left, that's the a ^2 -, 2. Once I factor out the a -, 3. This is a monomial now. Now we also want to see if this can actually factor any further, and it can't because this a ^2 -, 2 is not a difference of two squares. It wouldn't go further on the next page. I've given you a bunch to try and now you should be able to successfully complete them. Thank you and have a wonderful day.