Transcript for:
Factoring Concepts and Methods

okay so brand new thing for us we're going to talk about factoring and where we start with factoring we start with like the most basic idea of factoring really factoring polinomial by something called the GCF have you ever heard of the GCF before that stands for the factor that's right greatest common factor well like I said for some of you this is review we're going to go over what it means to be the greatest common factor what factors even are how to distinguish a common factor and then how to use them so the number one thing that we got to know is when we talking about the GCF is what in the world factors mean in the first place so when we talk about a factor or factors factors are these numbers so that and variables and or variables that when we multiply them together we get a term so when we talk about factors it's two or more numbers that multiply together to create a term so what that means is that most of our terms you can separate into component factors that can be multi together so two or more numbers or variables that are multiplied together to create a term one key thing I you need to know it's multiplied though uh we don't add two numbers as factors create a term we can get terms that way but the the addition of that's not what we're talking about we're talking about the multiplication idea and the reason why we're talking about that is because when we Factor we're really talking about dividing uh you remember this word called distribution yes remember distribution is the idea of multiplying and you get rid of parenthesis factoring is the opposite idea it's dividing to create parentheses we'll talk about that later so two or more numbers or variables that are multiplied to create a term that's a really long- winded way to explain something that that with an example I think it's going to be really clear let's do just a couple very quick examples here for instance 15 number 15 is a composite number it's not a prime which means it has some factors in there can you give me a factor of 15 three okay now factors always come pairs so if you give me three you also give me that's right so three and five are two factors that multiply together create 15 you guys are you all right with the idea of factors I'm not going to spend a lot more time on this but three and five are just two examples of factors of 15 now what some people get a little confused on is how in the world do we do this with can we do it with variables well the idea is anytime you can take two quantities whether they be numbers or variables and multiply them together well those things are factors so let me give you for instance here and then you'll give me another one X to 7 you all know how to pronounce it yeah say that for me perfect if I can find two quantities so two variables raised to another power that when I multiply them together they make x to the 7th then I found two factors for instance are you aware that x 3r * X 4th addition x x 3r * X 4th gives you x 7 you aware of that one then those are two factors of x 7 I want you to give me another one so give me two sets of uh two variables X2 whatever power you you can think of that would also make up x to the 7 because I have given you two but there's there's several examples we could do give me another one x s okay x s and x to the 5th sure as long as X2 and X to 5th multiply together to give me X 7th then those are two different factors of x 7 so fans if you feel okay with with this so far left side people are we okay with that so far how about um we do more and more how about if I give x what would the other Factor have to be perfect because this counts as x to the first power so if I have x and x to the six well you know what's interesting is when we look at this we have X X that's a factor x the 6 the fact x to the 2 the third the 4th to the 5ifth all those powers of X are inherently factors of x to the 7th because I can always break it down it's always being multiplied since X the 7th is just x * x * x * x * x * x * x * X I think I have one too many X's there uh we can always break off those x to the 2 * X the 5th those are inherently factors am I getting through to you guys on this one okay it's going to be important for our next thing now common factors so we've got factors now we're going to talk about common factors and after that greatest common factors and how to use them so what does it mean to be a common factor when I say the word common to you what what's that mean in in just in English what's a what's something common say what something have simar something they have similar sure something that have shared well that that's what it means here common kind of means shared so when we talk about common fact factors we're talking about shared factors I'll give you an example so if you want just a basic definition set of common factor think of it like a shared Factor you can write that down common factors me a shared Factor let me give you an example what we're going to do right here is we're going to list out all the factors that we can think of in 12 or 28 so all 12 and 28 so all those uh previous classes you had had like math 80 math 991 all that stuff where you you learned about factors well we're going to use that right now but when we get into the actual practice going be much quicker for us so right now we're going to list them let's give me some factors of 12 would you two and six you know they always come in pairs do you know they always come in pairs unless you have a perfect square like 25 all your factors come in pairs so typically what I tell students if you've had me before you know this start with one is one a factor of 12 one goes into any number right so one is always a factor one and one times what would give you 12 there we go so we got two factors already does two work is two a factor of 12 yes sir so we got two and uhhuh anything else four as soon as you cross over and get a factor you already have listed you know you're done for sure you don't have to keep checking so we have 1 12 cool we got 2 six cool we got three four how about four oh we already have it as soon as soon as you already have it you're done with your factors so as soon as you cross over and get one you already have you're done now let's do 28 what should we start with 28 yeah sure how about two is two a factor two and what 14 M how about three is three a factor no no does three go into 28 evenly no then it can't be a factor how about four does four work okay how about uh five how about six no no how about seven yes is it already on the board then we have listed all of our factors if you do it in pairs it's pretty quick like that not your head if you're you're with that so far now we're going to talk about the common factors all you got to do if you've listed them all out is look for the factors that are the same give me one common factor how about one one yeah does it does it do anything for no not really it's always going to be one uh two what else four four anything else okay so we know one's common two's common four not three not six not 12 they're they're not listed both numbers but the idea of these factors being shared is the idea of a common factor now what we're going to get to eventually is the greatest common factor we can probably talk about it right now what one of those factors is the biggest number that's the greatest common fact and that's the idea that you're going to be dealing with with a lot of man a lot of this chapter let talking about greatest common factor first now what's also interesting is we can list out the factors for variables and I alluded it I alluded to it over here let's talk about how you would list out the factors for 12 X2 and after that we're going to do 15 x to 4 it we talk specifically about how to how to deal with these variables well how about this one how about with the 12 do you realize that all of these factors 1 2 3 4 6 and 12 they're all still going to be factors of 12 x are you with me on that do you understand why why we don't have why we can list out 1 2 3 4 6 and 12 and even though this has an X squ these are still factors do you guys get that if you not if you're like well I get it but you really don't get it here's why um do you get that 12 x^2 is the same thing as 12 * x^2 if factors are these pieces that you get to multiply together to create a term then any factor of 12 when you multiply by something else will inherently create that that same term so anything that's a factor of 12 I'm multiplying it it's going to be a factor of 12x just by its very nature does that clear things up if you didn't quite get that so when we start with the the factors of a term that has variables you always start with just the factors of the numbers itself so something like 1 2 3 4 6 12 cool that's just the numbers now we talk up the variables so we're just going to T those on here think back over here when we had x to the 7 we had a lot of factors of x to the 7 in fact the factors were X X2 X to 3r X to 4th X to 5th x to the 6th one is already up there and x to the 7th so basically every Power listen carefully okay every Power leading up to this largest power is always a factor so when you list this out all you got to do is go okay well uh H is X to the first a factor yes is X to the second a factor is it here yeah X second is a factor is X the third a factor here no that'd be too much but what you're going to do if I listed this out okay I'm going to take a little shortcut if I listed out all the factors I would have one one goes into anything it's always a factor remember factors come in pairs usually so if I have one I would also have x to the 7th hey just just listen is it true that 1 * x 7 gives me x 7 those are factors multiplied together they have to be factors uh if if I do x to the just x to the first Power it's got to come in pairs what else that's right hey oh wait look we have that one uh how about x to the 2 can I multiply X second by something to get x 7 what is it x to the 3r yeah X 4th notice look at this what this says is if you do that by pairs which you said that we already can do right are are you with me if you do that then what you have listed out here is one and then every single power of X up until you get to that power so here we already have the one it's built in one's there all the time we have the two the three the four the 6 the 12 all by virtue of having that 12 in that term now we also have X is that a power underneath x s mhm x to the 2 yeah so we just list out every single power until we get to that one let's let's try it here okay you tell me what I'm going to have for my 15 what's the first number that I'm going to put up there probably one because I need the 15 so we deal with the numbers first uh what else three comes with a five is there anything else no so one and 15 two doesn't work three works with a five four doesn't work five we already get the five we're done we're done with our numbers now let's try the variables so I want I want us to list out what the variable are as far as our factors go what's the first one should I put x s up there sure is it a power under X to 4th then it's do you guys get that it's inherently a factor if it's less than that power because I can always write it as a power lower than this times another power x 3r is lower than x 7 multiply x 4 x the second is lower than x 7 multip x 5th x the 1st is lower than x 7 multiply X six you can always make that up that means that every single power below or equal to that one is going to be a factor we just list them makes it easy so x to the first yes x to the second yes how about x to the thir yeah of course how about x to the 4th yes yeah how about x to the 5th no no X 4th yes show hands and feel okay with the factors as far as variables go this part yeah probably pretty easy I know you know that the the variables are what I'm talking about right now that's what people get a little confused my question the one that threw me off was the 12 X2 at the very end where you have an X by itself and then the X squ CU When I see that I'm seeing three with the the two x's on the okay keep in mind that what's going on here is that we're we're listing out these terms we're not multiplying all these things together if you multiply all these things you're going to get way more than 12 oh gotcha okay and you're going to get way more than X2 we're just listing that if you multiply this all out wow that's x to the add all those things up what we're doing is saying well what could we divide by we could divide by one we could divide by x and x squ and x if you can divide by these things they are factors because that means that when you multiply two things together they'll give you that term so we're not multiplying these things we're listing them right now did that did that clear that up for you okay now let's talk about the common factors and this is where it's going to get important what are some common factors here what do you see one one three three great three anything else oh so as far as the numbers go anything else besides one and three what's the listen carefully okay this is where we're going to jump to the greatest common factor what's the biggest number good now we look at the variables almost independently so uh do we have common factors yeah this is weird though right every Power is going to be listed up to the the largest one so X is a common factor what else is a common factor now how about X 3r is that a common factor is it in both places no how about X to 4th no no okay what's bigger X or x s same question as what's bigger one or three so when we get to talk about the greatest common factor here's what you do and listen carefully you find the greatest common factor of the numbers that would be the three you with me you find the greatest greatest common factor of the variables what's bigger you put those two together so the three and the why not the X biggest why not the X to the 4th not it's not there you take the biggest one that's shared by both that by definition is the greatest common shared Factor so what we do we do the numbers we do the variables we put them together this right here is the idea of the GCF what I want to know if is that does that make sense to you show fans if it does you feel okay with that yes yes yes Destiny yes okay cool kind of cheated and went a little little bit ahead so when we're talking about this we know what factor is factors are those pieces that are multiplied together give as terms all about multiplication common factors when they're shared and now we've jumped to the greatest common factor the greatest common factor is the biggest shared Factor so when we say GCF what we mean here is the biggest Shar fact and right now what I'd like to do I'd like to just go over some examples and and get us really really really good at finding this if we fail at finding GCF we're going to fail at factoring this is huge for us it's the very first step of your time is factoring the GCF whenever we talk about that getting through to you so let's practice we're not going to even worry about variables right now we're just going to do some numbers just find in GCF um are there any questions on factors common factors or the idea of the GCF I know this is not purely explained yet uh but I want to know if you understand the idea that the greatest comp factor is just this biggest thing that's shared by both these terms do you understand that biggest Factor okay good it's shared by both terms all right well if we're going to talk about the greatest common factor this bigest shared Factor we got to have at least two numbers so let's start with 45 and 75 yeah it looks like a 75 to me doesn't what I said oh jeeez come on 75 I'm going to give you two ways to do this um the first way is the way that you already know I'm going to go very fast through that that method one this listen them all out that's your first method you should know how to do that from a previous class the second method some teachers gloss over so I'll make sure you at least see it once the first method is just list them all out so if you have trouble with finding the GCF just in your head by the way we need the greatest so I don't want you to just give me one all the time what's the greatest common factor one no that's a factor is it the biggest one generally not unless you have two relatively prime numbers um but no so if we're going to list these out we got one and we got 45 we'd have three and we'd have 15 we'd have five and we'd have nine did I miss any am I good can you think of any more I can't think of anymore so 135 9 15 45 I think I have them all let's do 75 again I my job right now is not to teach you how to do this this should be there already my job is to teach you what to do with it uh afterwards you follow so the next one uh we got one we got 75 what else 3 and 25 yep 3 and 25 perfect what else 5 15 okay anything else can you think of any more than that that's all I have written down okay uh well do we have any common factors what's the first one one is it the biggest one no are you always going to have one of course you are uh three is that the biggest one no five no tell you what if three and five are both here that means you have something bigger because if they're both multiplied to give you 15 you're going to have a 15 15 is that the biggest one so right now we know our GCF is 15 for small numbers that's probably the way to go I will give you another way to do this not hard just a little bit more kind of advanced thinking uh use some number Theory here so if I had 45 another way we could do this you could do this with a number tree uh number trees with a with factor trees with greatest common factors and prime factorization says you start with the smallest prime numbers and you break it down until you have only prime numbers here for instance 3^2 * 5 would be the prime factorization if you did 75 it start off really similar you'd have 3 * 25 and that would give you 5 * 5 so we would have 3 * 5 2 I want to know if that makes sense to you have you seen that before yes might not remember exactly how to do it uh but you've seen it before you start with a smallest primes not the biggest one so you start with two and if there's no twos you go to three then you go to Five you seven nine sorry not nine 11 then 13 and so forth and so on well if we did this we got 3^ 2 * 5 we got 3 * 5 2 the GCF will always always always be the smallest powers of every I hope some of you guys are Z I hope you're not zoning out some of you guys are are not listening right now this is a really good thing to do with large numbers tell you right that it's probably the best way way to go take the smallest power of every different Factor the factors that I see here are three and five and then three and five again do you see that you can always find the GCF by doing this uh what's smaller folks smaller 3 squ or three so GCF should be yeah I'm sorry which one three that's smaller times what's smaller five or 5 squ what's 3 * 5 always always works uh why that's nice is because if I give you really big numbers like uh let's do the greatest common fact of 196 2000 187 or 84 you want to do that in your head you're going to want to write out every single factor of two whatever I said know I just made it up you going to want to do that no but prime factorization is pretty quick that's another way that you can go okay let's move on just a little bit how about uh how about 32 and 33 32 and 33 32 and 33 what the what are the factors of uh 32 1 and 32 I'll give you that one I'll do the hard one for you okay 1 and 32 I got it got your back 2 and 16 give me another one someone else four and eight four and eight how about five six no seven eight I already got it so we're done uh 33 is going to be a little bit quicker here we got one and we got 33 anything else four doesn't work five doesn't work six doesn't work seven doesn't work eight doesn't work n doesn't work 10 doesn't work 11 does but it's already up there so we're done here's my question are there any common factors any common factors one there's always at least one common factor and it's one so are there cases where you can have two numbers which only have a greatest common factor of one yeah it's called if you want a fancy word it's called relatively prime means that they have no common factors besides one nothing will be common that you can factor out of those two numbers this right here this is what we're looking to have after we've factored is numbers that are relatively prime that no other number goes into it besides one so here when you have a GCF of one it means that if I have two terms with those numbers I'm not going to be able to factor it the way that I want to how about if I have uh oh let's see how about 14 24 and 60 are we able to find the greatest common factor of three numbers is that possible it it means the same thing it just means hey let's find the biggest number that will go into all three of these so you can do it the same way that you did before you can do it this way you can list out all the factors if you want to and find them just find them so one if you want to do it kind of quickly if you want to do it quickly and not write everything down I'll give you a little helpful hint here start with the one that has the least number of factors for instance um if I was going to do this in my head and I didn't want to write all of them down because I don't like doing that okay uh I'd start probably with 14 and here's what I'd do I'd say 14 I'd write them out so I don't miss any you you don't want to miss any I'd have one I'd have 14 I'd have two and I'd have seven would we agree those are all the factors yes okay to be a greatest common factor of three numbers do you do you get that they have to go into all three of them yes so if I'm uh let's see like here six goes into both 24 and 60 doesn't it but does it go into 14 it can't be the GCF it's got to go into all three so if you pick the number with the least number of factors sometimes you can cheat a little bit one do I care about one no no okay two does two go to 14 obviously does two go to 24 does two go to 60 potentially that's it check the other ones just to be sure that there's nothing bigger does seven go into 24 no s goes into 21 it can't go into 24 does s go into 24 no do we care if it goes into 60 once we've determined doesn't go 24 so this doesn't matter does 14 go into 24 no you have quickly found the GCF just by the process of elimination pick the one pick the number that has the least number of factors just check the factors if they don't go into both numbers then they're not the GCF show hands if that made sense to you could you have done this way and listed them all absolutely but I'm trying to save you guys some time I'm saving your lives right lives are time I'm saving your lives so the GCF for these ones just two very quick way to do [Music] it okay I kind of previewed this a little bit in the example that was right here before we're going to talk about variables and then we're going to put all this stuff together and learn how to factor are you ready this is like this is the good stuff so here we go next example let's talk about the variables if I wanted to find the GCF of X2 X to 4th and X 5th GCF of x^2 x 4th and X 5th for those of you who get this idea already this is a really easy question really easy if you don't I'm going to make it easy right now let's do let's do this with what we have here so if I were to list out the factors you remember list out fact you remember you remember you remember remember listen out factors how when I do this well there's always a one but then I'm just going to have all the powers leading up to my largest power that's what I'm going to have let's look at this real careful do you understand that one's always a factor always divides everything is X a factor it's a power of X less than two is X squ a factor of course it is because one times that gives me this this term if I'm going to do X to 4th well I'm just going to have one but I'm have X and X2 and x 3r and x 4th every Power leading up to X 4th if I do X the 5th I'm going to have 1 and x and X2 and x 3r and x 4th and X to 5th I hope so did I do it right okay now here's my question do we have any shared factors yes is it X to 5th tell you what a lot of students when they're first learning this they look at that and go oh yeah X 5th that's the biggest number it's the biggest one obviously that's going to be it it's greatest no greatest Shar so all these all of these have at least x s correct well that's really what we're looking for we're looking for all of them have at least this much if you look here the biggest one that's actually shared is right there I don't care that that's x to the thir and it's shared there I don't care that doesn't have it so it's not the greatest common common means shared the greatest common factor that's what we're looking for show fans if feel okay with with that idea cool so in our case man it becomes really easy you're looking at these ones well I I know it seems funny but take the smallest power if you take the smallest power that's listed right here that's the shared that's the one that's shared you can't do X 5th because that doesn't have X 5th it has most x s does this have this has x s yeah does this have at least X squ does this have at least x2 in it and that's your GCF so GCF here would be X2 I want you to try one just real quick to make sure that you you all get the hang of it let's do um a couple examples what I want from you is I want the GCF of Y 4th y 5th and Y 8th and then I want the GCF of X and X to the 10th and you can just write it right next to this if you need to list out all the factors like I did here do it until it really Mak sense don't do shortcuts right now because this is imperative you understand this so if you need to list them out all list every Power of X up until that number list every Power of X up until that number list every Power of X up until that number look for the biggest one that's your GCF you can do that every time once you understand this concept though you can go very very fast on that does anyone need more time on these have you figured out the GCF for y the 4th y the 5ifth and Y the e8th what's the biggest power that but that's shared by all three of those terms what is it perfect that's right you can't go any bigger because that doesn't have any more than y the 4th uh it's kind of it's kind of like going out to lunch with your buddies and you want to make sure that everybody eats and uh you have $8 and you have $5 and po Leonard has $4 and you go let's go to uh let's go to this place where it costs $5 can I eat at $4 can I eat no let's go this place has $8 can I eat screw you guys sucks uh where where's everyone going to be able to eat at the $4 restaurant that's my type of folks you got to go where everyone can eat so go where the power is shared by all y the fourth hopefully that analogy Mak sense to you how about x and x to the oh wow how many X's are here what power of X is that one x to the 10th what's what's the greatest common factor x you can't go any bigger than that because this guy only has $1 poor guy uh that's it it's that has more sure but you need to go with the largest factor that is shared by all the terms that are listed have I explained it well enough you guys understand it now we're going to put the idea together we're going to talk about the greatest common factor of these terms that involve both numbers and variables but it's no different than what you've done did you guys get the numbers did you get the variables do the same thing you should put them together at the end so let's talk about that real quick did you know about that by the way if you didn't is that kind of neat works every time it's really awesome I like it so we're going to find the GCF here I'm going to stop just writing out all the factors we're going to get to the point where this is going to be very easy for us to do in our heads because we're so good at at at uh doing factors of numbers and listing factors in our heads and variables now so we're going to stop listing out everything it's going to take way too long and the idea is get the greatest common factor quickly and then then use that to factor we're going to be doing that in just a minute so um what I want you to do here when you're factoring out the greatest common factor when you're looking for the greatest common factor do the numbers first then do the variables and just combine them so I remember that we we had an example like that right it kind of cheated and went in advance I'll do that from time to time let's look at the numbers the numbers here are what's the what are the coefficients is what they're called what's the coefficients 9 is one of them someone else what's another coefficient 15 and then is there a number that goes into all three of those numbers you can always answer yes because it's one I want the biggest number that goes into all three three three does sure so one's not going to help us six doesn't work so when we write out GCF we can do it in a two-part process we do the numbers first so three it's true right three goes in 9 15 and six we're good and then we do the variable so when we're looking at the variables it's basically doing this process first exactly the same just like we got the two we got the three do the same thing then do the variables just like we got the X squ here or the X what's the greatest common factor as it refers to the variables in this case what is it X that's right that's got more that's got more but well that's the that's the smallest one so we have to go with the guys got the smallest amount of money here this guy's only got a buck so yeah these guys have at least one X but we're going with the greatest common that has to be shared so not can list out x to the 4th that would take this guy out of the picture and that guy out of the picture xra squar would this guy would have it but that guy wouldn't so x to the first power is as far as our variable is concerned gra common factor what that means is when we put our numbers and a variables together greatest common factor is 3 does 3x go into this term sure does remember goes in means divides because a factor is multiplication so the inverse of that is division does 3x divide 15 x to the 4th of course does it divide 6X yes yeah notice how if this was a square would x s divide x no too big too big if that was a four would that divide too big it's a problem I think I said four and wrote two didn't I oh well no you said two I did you said two first for two then WR four I got to write yes yes perfect tell you what I'm going to do one more with you kind of a more advanced one looks more advanced not harder it's same exact thing then I'm going to have you do a couple on your own okie dokie so you can practice you so first things lot's going on it can look intimidating but follow the explanation here follow the explanation the explanation is find the the GCF for your numbers and then look for the GCF of your variables don't just do it all at once it can be kind of hard to do so first thing let's just see let's pretend there's no variables let's look at numbers what are the coefficients here coefficient 11 it's not it's never nothing one one yeah if there's nothing there is one uh so coefficient 11 1 hey what number goes into all three 11 1 and one one do you need to write down the one no really I prefer if you didn't but I'm going to put it here just to say that we did it okay uh now we're going to go over to the A's and our B's you know what just like we can do our numbers independently of our varibles you can do each variable independently of themselves because it's all being multiplied together so don't worry about the B's right now look at the A's I got a to what one do you know that you're done right now yeah a to the second and a okay what's in common to all three of those not a to the second not a to the 4th cuz I have 1 2 3 4 not that no no don't do that okay you're looking at these independently saying what's the biggest value of a power of a that's going to go into all three and that would be a to the first are you with me on this one are you sure yeah guys in the middle yes no okay we're all good so one goes into all the coefficients a to the first Power well that's the greatest common factors as far as a let's do the B I have B 3 I have B 3 and I have B squ what's the biggest one that goes into all of them perfect erase this one that is our GCF does it make you know what you should always check it too you should always check it does a b^ 2 divide this in fact if I divided it I would have B to the first Power left over are you with me on that one yeah does this divide this I would have a * B if I divided that does this divide this I would have what one one are you ready to try some on your own yes y easy medium hard Med what do you think Medium you're still learning it right is it going to take practice bet you it's going to take some practice do the numbers do the variables you can do each variable independently just don't start picking the biggest power that you see that's a problem pick the biggest power that's shared by all of them and you get it right every time I hope you understand that doing this doing this is so important for what we're going to do next because you can't Factor unless you know how to find the greatest common factor I mean it's it's right there you got to know how to do it I think you'll be fine at it but you will need to practice it Okay so how about how about that I want the GCF -8 y^2 -63 y 3r 27 y 4 one thing I want you to look at here uh did you notice how that negative had nothing to do with what that was it's interesting right okay we'll talk more about how to use that to your advantage when we actually into factoring so I want the GCF here I want the GCF here and I'll give you one more just for funsies actually some of you might be done just after writing them down huh they can go pretty quickly oh man let's look at the first example have I given you enough time to to complete these or at least do the first two all right first example what should we do should we look at all of this or just the numbers first numers do numbers 18 do I need to worry about the negatives no when we're fact when we're actually factoring I'll show you how to do that appropriately but right now no and and it's not going to affect the actual absolute value of what you factor anyway so 18 6327 oh man have you found something that goes in nine goes into all of you know you could you could check it with maybe this one or maybe this one because it has a least number of factors this say 1 3 9 and 27 does 27 work here no no maybe go do the nine does nine work here yeah does nine work here hey you're done it's nine now variables at this point should be really easy it's easy it looks I mean you think it's going to be harder it's not the numbers are harder what's the greatest common factor as far as variable is concerned that are shared by all three what is it y perfect that's exactly right you don't add them all up and get y to the 9th or something you just get well the one that shared by y s show hands we got 9 y^2 brilliant very well done okay how about the next one uh numbers first we got four we got we got three your already uh what's the number as far as your GCF one you can write down the one just to keep it in our head that we didn't numbers uh now the variables have X2 X 3r X 8th what is the greatest common variable so we're going to have 1 x^2 erase the one you don't need it but let's double check does x s go into this yes yeah does X squ go to this yeah go to this yeah good did you get to that one kind of fun right what's the uh what's the greatest common factor as far as the numbers concerned one there there are no coefficients besides one so you're going to have a one you don't even need to write it if you don't want to let's talk about the A's remember you can do the A's and the B's or any variables independently of each other so look at the A's first what's the greatest common factor in reference to the in regards to the powers of a why is it a s and not a to 4th okay so even yeah so even though a to four is the biggest power of a it's not the biggest power of a that can be found in each term a the second is how about the B's sure does that have at least this in it you can divide right have we gone over any of the powers no okay so we're good that's our greatest common factor we got that one perfect we're going to go just over a couple examples about how to use this really I want to just get the steps in your head that way next time we come in this is going to be just we're going to be money on this do you feel okay about uh finding the factors of numbers variables putting them together if you can do this you can factor and here's how we do it by the way uh what I'll do sometimes when I first introduce something I'll give you an example and I'll give you the steps on how to do it so you can refer back to how to do this later so let's start with something very simple uh some of you might have I'm guessing you have seen this before let's do something like 6X + 14 6X + 14 now instead of just listing out the factors we're going to be factoring we're going to be using that greatest common factor finding the greatest common factor to our advantage now here's a deal you already know what distribution is correct yes so if I gave you you don't need to write this down just just watch I'm erasing them in any if I give you 5 * x + 2 I said distribute you would multiply this here you would multiply this here and you get 5 x + 10 yes no what we're doing in factoring is literally going opposite of that in Reverse of that so whereas distribution is multiplication to get rid of parenthesis okay focus up here for a second whereas distribution is multiplication to get rid of parentheses factoring is the opposite what's the opposite of multiplication and that means we're going to be creating parentheses so when you factor you're basically going not from here to here what you're used to you're going from here to here you're going in reverse so we're not multiplying we're dividing we're not getting rid of parentheses we're creating them quick head out if you feel okay with the idea of that good so step number one all the stuff we've been doing is Step number one so find the GCF of all the terms um I am expecting that you understand what a term is when we look at this term term not term term term terms are those pieces of a mathematical expression that are separated by pluses and minuses so when you look at this problem this uh expression this is one term that's one term you all that one okay so what we're going to do first find the GCF of all the terms well let's do that let's do the GCF alls tell me GCF can you look at 6X and 14 just pretend it's like this okay pretend it's like that can you look at 6X and 14 and tell me a greatest common factor what about variables are there any variables that are shared no ah so it's not like this or this where we have shared variables this has an X this don't have an X it can't be in the GCF what what you say the GCF was again two two two is the biggest factor that goes into both these show be okay with that the number came first we had no variables so we're good step number two what you're going to do with step number two after you found the GCF write the GCF in front of some parentheses so step number two says uh I'm sorry one more time what's our GCF and factoring it gets rid of parentheses or it creates parentheses so we're going to do our GCF and then we're going to put a set of parentheses this is what I mean by step number two so step number one find GCF oh my gosh we should be so good at that right now right we spent an entire hour Finding GCF now write the GCF in front of some parentheses because that's what we're creating here after that step number three distribution multiplies factoring divide each term we know what terms are divide each term by G CF just divide that's it you all know how to divide even if there was variables you would know how to divide that we'll show we'll see that in some subsequent examples but here we go so step one find GCF you all feel okay with GCF step two write the GCF hey already done got it step three just divide take each one of these terms and divide it by the number you've just written so can you do 6X / two what's 6X / 2 3 it still has the x correct cuz there was no X to divide so okay plus hey plus uh can you do 14 / two that ladies and gentlemen is Factor step number four just check your work make sure you have it right do not go any further until you have it right here's how you check okay listen here's how you check whether you have it right the first thing you do you distribute to make sure this is accurate is 2 * 3x 6X is 2 * 7 14 the next thing you do you make absolutely certain that you have found the GCF what that means is that the remember the special word I gave you phrase It's called relatively prime remember that that these numbers have to be relatively prime if they're not you've made a mistake can you divide three by and seven by anything besides one no that's good are there any shared variables here no that means you factored the GCF correctly and that's the whole idea so step number four if you want to write step four it's just check your work and that's the two ways that you do it show hands if today has made sense for you good okay so we're going to continue factoring and the very first I'm going to say this probably 100 times the very first step in factoring anything that you're going to factor is always finding the GCF if there is one that is your first step no matter what else I teach you for the rest of the semester whenever you're involved in factoring very first thing you check for is what GCF that's number one so in our case up here the idea is I'll Le you through this because it's been a while since we've done any of this the very first idea is look at how many terms you have remember terms are those things those pieces of a math expr expression that are separated by pluses and or minuses so pluses or how many terms do I have two we look at those two terms and basically we just find the GCF like we practiced so many times before with those two terms remember that we're looking for numbers first and then variables and we put those two things together so when we look at 49x - 35 those two terms 49x and the 35 we think of a number any number that that can go into both of those terms and the biggest number the greatest number that goes into factors both of those terms common greatest common factor what number goes into both of those okay so I'm not going to rewrite the steps I gave to you last time but the idea is once you have determined what the GCF is you can write it down you don't really have to but you can just keep in your mind we get the number first then we look for any variables are there any variables common to both of these terms so while this has an X that one doesn't we we say no there's no variable that's common to both hence our GCF for for this expression is just seven show F feel okay with that so far cool deal now the next step I gave you was well factoring is the opposite the inverse of distribution so where distribution multiplies and gets rid of parentheses factoring divides and creates them so we're really creating parentheses every time we factor that always if you factor something you're going to create parenthesis so when we've determined our GCF is seven what I've said to do is put that seven the GCF right out front and then create a set of parentheses what you're saying here is I'm going the opposite of distribution uh I'm going to create parenthesis and divide to get my inner two terms here so basically since distributions multiply factoring is divide to get these two terms we're just going to divide what are we going to divide by why that's it that's a GCF that's what actually divides into both of those terms so we're going to take that sometimes you can do them in your head when they get more advanced I'll show you some methods on how to write this out so that you can you can see really what's going on basically you do it term by term can you do 49x / 7 what is it that goes right here that's our very first term so Bas it's just these two terms divided by our GCF and we put our remaining two terms inside our parenthesis can you check your work yeah if factoring is the opposite distribution that makes a checker work we just have to distribute so 7 * 7 x is 49x we know we got it right also this sign is going to be the same unless we start factoring out negatives and we're going to talk about that in a little bit but generally if you're factoring out positive numbers your signs don't change now we go to the next one what's 35 remember we're dividing we're factoring what's 35 ID 7 5 and you always double check your work don't go any further it be so silly to go further cuz right now this is the problem right right it's can you factor or not later on in chapter 7 8 9 I not 9 but 10 11 12 13 you're going to be factoring but that's a small part of the puzzle so don't just assume you have it right and go on and get the whole problem wrong because you'll waste a lot of time there make sure you got it right so first thing we check for does this work is 7 * 7x 49 7 * we can consider it Nega 5 I know it's minus we consider 7 * 5 is that the minus 35 that we want yes cool are these there was a key word special words relatively prime are they relatively prime which means the only thing that goes into both those terms is one then we're done that's the idea of factoring do you guys feel okay with the factor I know it's been a while but does that make sense to you okay so that's our factoring let's go ahead and do a couple more examples we're going to start ramping it up uh this was probably the easiest one that we can do at this point uh we're going to start talking about how variables play a part of this and then um then put numbers variables together and see what all we can do so next example how about y 5th - Y 7th y 5th - Y 7th uh first thing how many terms do we have what do you think perfect what's the first step in factoring every single time what is it g let's look for the GCF and for now we're going to write it down when you get uh more practice with this you're not going to do that you're just going to go for it you're just going to factor so we're looking for the GCF um are there any numbers which go into those two terms one one yeah so just one do we have to write down the one no you can but it's not going to help us so next we look for any variables are there any variables which go into both of those terms remember it's very much like doing this and saying hey what's the GCF of those two terms you're doing the same thing it's just now we have a minus between it but this idea does not change what is the GCF according to the Y the 5th and Y the 7th remember that what it is it's the biggest power that both of these things have not the biggest power you see but the biggest power that both of these terms contain what is it that's right feel all right with the GCF hey we're factoring what what do I do with that y the 5th where going I put that M so if we have y the 5th and we know wait a minute factoring means that we're going the opposite distribution so we're going to be creating some parentheses here that always happens when you're factoring now we're just going to figure out what these two terms are going to be you still have the same number of terms don't you inside those parentheses there's two here there's going to be two terms here you're just creating a factor something that you can multiply by that gives you back your original expression that's the idea now if you're a little bit kind of like oh my gosh I don't know how to divide with these exponents here's a TR not a trick um something that you can do to help you out in order to find these you guys all all know that you divide right well take your first term divide divide by your GCF and whatever your answer is by the way this has to work otherwise you picked the wrong GCF it must always divide otherwise it's not a factor did you catch that if you're trying to divide and go wait a minute it doesn't work pick the wrong thing so go back and fix your GCF this will always work how much is y 5id y 5 that goes here one you can always do it off the side if it gets a little bit more advanced for you man I can't do in my head write it out not a big deal is that sign going to change what do you think next one this one so we wrote the first term ID GCF that's y 5th / just happened to be y 5th it ends up being one that's our first term now we're going to take Y 7th we already know that we're going to have a minus there divided by y to the wait divided by what sure second term divided by GCF you have two places can you do Y 7th / y 5th yeah y 7 how do we do that what do you do with those those numbers there we know that when we're dividing expon or common bases with with exponents we subtract those exponents so we have y second and that Y the second goes there can you still check your work even though you dealt only with variables here yeah yeah distribution is the opposite of factoring so those things work hand in hand like Division and multiplication it is division and multiplication so if we checked it y 5 * 1 y 5th and Y 5 * y number oh yeah exponent rules when I multiply common bases I add add those Powers so that would be y 7 and we know we have it right now let's keep going just a little bit further um how about this is there anything goes into both one and y^2 can't let's check for that is there anything goes into both those things one one does anything bigger than one are they relatively prime yes and we double check so we know we're good show P feel okay with what we're talking about okay now in the future we will be talking about how we can go further on something like this uh this if you if you know this concept if you had this class before or that factoring before which I think most of you have remember we it's kind of like a a refresher for you that's called the difference of squares and we're going to talk about did you know that term that difference of squares term yeah if you remember it great if not we're going to cover it that's called the difference of squares and I'll show you what to do with that as we we move forward okay what I'd like you to do now I'm just going to give you two examples I want you to do them on your own to make sure that you get this concept when I do this they going to be really really similar than the last two that I gave you it's just so you can have that that practice uh so that I don't want you completely F your notes if you have to have your notes okay but try to do this fresh without having to go back and that way you you you actually get it embedded in your head you guys with me on this stuff you're quiet today are you okay yeah I'm excited so might be that I'm extremely tired and running just purely off adrenaline but whatever people on the video don't know this but had a baby in the case not me personally weird but uh anyway go ahead with these two so we're just working on factoring the GCF so I want you to do that one these two so the idea is factoring is the opposite of distribution instead of multiplying we are dividing dividing so the first thing we have to find is what will divide all of our terms remember it's all of the terms here in our case we only have two in the next examples we're going to have three four so forth and so on so for us in the first example we're looking at 4 T ^2 + we got to find a number or variable or both that will divide both of those terms what number is that four so if you want to write GCF great if not that's okay all you got to do is write the GCF whatever you have in front of some parentheses factored means we're creating parentheses and then start dividing what's 4 t^2 first term divided by our GCF so 4 t^2 divid it by four you even think of it like that what is the answer our first term inside of our parentheses is the plus going to change okay 12 / 4 or just yeah have you double checked it to make sure that you're right don't go any further before you double check it you got to make sure okay I I don't seem silly right a pretty easy one yeah okay it's easy but later on when they become more difficult double check it can we go any further here is there a number or variable that goes into both those terms no no relatively prime is that key word we're good to go we can double check the distribution our head next up tell me the GCF I just want the right side people these are two terms tell me the GCF with Y the 8 plus y 5th what's the GCF there cool that's the largest power of Y that is common to both of them y to the a it's too much for this guy did I give you the lunch an going to okay y the E if you need to write it down maybe write it real soft there divide y 5th what's y 8 / y 5th plus plus okay cool and now y 5th / y 5th what is that show hands if you got those two that's great that's fantastic that's factoring the GCF are you ready to step it up just a little bit m by the way in the previous examples this one and this one this is called a difference of Cubes I'm going to show you how to do that later this right here is called sorry difference of squares this right here is called a sum of cubes and we're going to talk about how to factor this you can actually keep going on this problem right now as far as a GCF you're done but we can do other kind of brands of factoring as we go through these problems so let's start getting towards that oh dear all right so -12x 3r we get a + 10 x^2 - 2x uh left side people real quick how many terms do you have so if we're going to factor by the GCF which is always our first step we've got to have a GCF that goes into just two terms right three terms all three terms all three terms so in our idea for finding a factor by the GCF you got to come up with something some factor that goes into all the terms not just one or two but all three of them or all four if you have four everything so let's look at the numbers first remember we can do numbers and variables separately let's look at the numbers I got -2 I got 10 and I got a -2 for my coefficients can you tell me a number that goes into all three of those remember that it doesn't have to be negative or positive for it to have a factor so -2 doesn't really matter what number you going to pick cool two goes into 12 10 mhm two cool that means that we can divide all three of these numbers by two is that true that means we've been factored by it is that the biggest number that goes into all three of them we want the biggest one that's the greatest idea now we look for the variables remember we do them separately so I got X cubed I got X2 and I got X can you give me a variable that goes to all three of those factors all three of those terms yeah X if we put that together that right there is our GCF okay uh middle people tell me the next step what am I going to do put it outside the parth okay so I got my 2x I know factoring means parentheses quick question how many terms are going to be inside the parentheses does this get rid of terms so if you have two terms you're going to have two terms and a factor if you have three terms you can have three terms and a factor it's actually making it look a little bit bigger isn't it so inside there's going to be the same number of terms but outside you're going to have this factor that you've divided out now we start dividing if you need to do this off to the side do it there's no sense in trying to do this in your head and getting it wrong if you have trouble dividing this by this and that's what factoring is just write it out there -12x / 2x that's that's fine I don't mind seeing well I do mind that one why if you do that then sure that's okay how much is -12 / 26 be careful what is it62 and yeah X Cub / X remember this is X to the 1st so if you subtract them 3 - 4 is X that is your very first term if you want to write this off to the side do it if it helps you do it if it doesn't if you can do straight in your head that's that's fine too I don't care you need to know how you guys learn so if you struggle with this now I'm always getting the wrong answer write it out if not if you're if you quick it makes perfect sense I can do it in my head cool do it in your head let's do the next one I'm going to do this one just in our heads here how much is 10 x^2 ID because remember we're factoring divided 2x so 10id 2 gives us and X2 / X gives us plus or minus lastly plus or minus here and then oh yeah that's that's a nice one how much is 2x ID 2x notice how when you're factoring you don't get rid of terms they can change to ones but that's the lowest that you can get right there so when you have a GCF that matches up with one of your terms you don't get zero you get one because something divided by itself is always one that's the idea show right with that one okay now can you check it should you check it should you check to see that these are all relatively prime mhm don't go any further before that now I'd like to do something else a lot of times in math it's really really helpful to factor so that two things happen one that your polom this is just a math expression inside here with more than one term uh that your polinomial is in order in order means that it goes from largest to smallest exponent ending with a constant so for instance this out of order you see what I'm talking about this is out of order this is in order this is in order in order in order do you guys get it this is in order we Factor so that that happens we like to put things in order even before we Factor so here we probably would have done this a little different second thing we also like to factor so that the first term of the polinomial that is in order I'll say that again because some of you guys are zoning out already don't zone out Z out you you want to factor so that firstly it's in order you get it secondly that the first term is positive that helps a lot because the next techniques we're going to do are dealing with these guys how you factor this and it's really really hard if that is negative does that make sense to you so when you do this process when you factor out your GCF you could have one of two answers one could be just like this 2X and this is this is perfectly fine as far as factoring the GCF but there is an alternative and this is the one that I'm going to prefer because it makes the rest of our lives easier instead of instead of going just with 2x check this out follow follow the logic here um is this in order from largest to smallest as far as the exponents go yes that's great that's what we want 3 2 1 you with me okay when I factored I factored 2x did I get a positive here this is what I want to be positive how can I change this to be positive what would I have to do to my GCF okay so instead of having the GCF of 2x let's say I did a GCF of -2X is it possible yes will -2X still divide that yeah and that yeah and that yeah what's going to happen to all three of my signs they're going to change that's the idea so you have Choice here okay the both of these are going to be absolutely correct but as far as continuing the proc process of factoring it makes it a lot easier if you factor so that your first term becomes positive the one thing you got to worry about is that you don't mess up the rest of your sign so I'm going to lead you through it one time real quick and then we're going to continue with another problem uh if we factor out the -2X tell me something what's uh -12x 3r / -2X remember a negative / a negative is a band so we got our 6X s posi 10 X positive / a negative is a negative / a negative is a are these the same yeah these are well do they look the same no do they represent the same thing yes yeah which one's going to be nicer for us it kind of depends on the context of your problem this technically is factoring out GCF so is that one this one would be easier to work on after this though to fany if you understand the difference there okay cool tell you what I want you to try a couple of them on your own in fact what I want out of you I'm going to give you two of them that I know that you can do really well and I'm going to give you one that's going to challenge you just maybe a little bit see if you remember how to do this are you guys ready for that question um you know how you said like when you're check and make sure everything's relatively relatively Bri what happens if it's not like so correct if you if you have some terms like two or three terms that are not relatively Prime you did this wrong factor and GCF will create if you do the GCF not just a CF CF is common factor right if you do the greatest common factor your terms will be relatively fine they have to be otherwise you would have something else that would divide all those terms and that would be part of your GCF does that make sense which is why I'm always having to check I don't want you to just factor out uh why here because you factor out why these things are not relatively Prim but um well never mind no cuz I was going to say six isn't really relativ relativity Prime with what cuz six can be divided by okay what relatively prime means is that when you look at this number and this number and this number nothing divides all three of those numbers except for one it's not just that the S I mean yeah you're going to have some numbers that uh you can Factor the numbers individually but what Rel relatively prime means as this number relates to a different number only one divides it that's what relatively prime means not that you can't uh decompose it as a composite number that it's kind of a different and I makeing my point that those are a little bit different of an idea so here's what I want for me I want you to do the first two problems I'm going to say that you're going to be just fine on these because they things like what we've done the next the last one I'm going to give you a little bit challenging I know you can all do it but pay attention to what the actual GCF is are you guys ready for this okay so first one I want you to identify the GCF this is up to you you can use a positive or negative I don't care I would you know how I would probably do this problem so pick the correct GCF and Factor it and then after that I want you to do this one this another one that I think you'll have no problem with this last one may be a little bit challenging I know you can do it okay there you go I'll be walking around in just a minute if you need help on this now is the time to ask don't wait until it's too late also keep in mind that if you want to really learn how to do something without relying on something else as a crutch try to do these without using your notes verbatim okay try not to go oh what's he doing okay now let me follow that step if you have to at this point do it okay because I'd rather you get that at least that down but if you can do it without your notes that's better that's what I want you to do as you're doing your homework don't just always go back and forth between notes and do an example that that's that's lame okay because when you get your test it's not going to be embedded in your head so get to that point where you can do these problems without notes and you are going to be absolutely golden as far as factoring GCA for I know right fractions are you are you fraction kidding me here fraction no just Kidd uh yeah fractions already seriously hey by the way can you uh can you check your work on all these problems yes do you have an excuse if you get these wrong no not really because you can always check them just by distribution remember distribution and factoring are inverses so all it takes a little bit of multiplication get rid of the parentheses and you should know if you have that right okay if you're still working great have you guys finished at least the first two of them okay I'm going start on the first two up here so first one oh my gosh -8 y 6 + 16 y 4 - 8 y^2 can you give me a number which goes into all three of those terms eight does did y all get eight we don't want to just have two we don't want four we want the largest the greatest common factor so for us we're going to factor out at least an eight uh now how about the variables are there any variables which are shared y the yeah don't just get the numbers we got to have everything here so numbers great eight variables y the 6 y 4 y^2 well the biggest one the greatest one that goes into all of them is y^2 y 4th is too much CU this guy doesn't have it why the eth is why the six is way too much cuz both of these guys don't have it you wouldn't be able to divide by that and if you if you don't believe me check this out if you did 8 y 6 as your GCF and started trying to divide these things is that going to work no that's too small for that to to work uh that means that when you start factoring like I showed you those steps before you come to an impa and that's not a good thing we don't want negative exponents of pols right now okay that's really awkward so we got 8 y^2 and then we have how many terms are going to be inside three you know what there's one thing I want to talk about right now is this in order okay might I choose something just a little bit different than this the Eight's perfect the Y squ is perfect but I might choose something a little different what might I do instead yeah I might do that and the reason is what is the reason again to be positive yeah if this is positive that helps us a lot for future factoring and for solving things and so typically is it is it a must right now no the must for you to understand is where in the world that 8 comes from where in the world that y^2 comes from and how to use it to get these three terms that's what you have to understand in the future we're going to be doing this because it's going to be really timesaving it's going to be really useful for us so this is what you have to have if you get this right now that's great if you get that well we're going to pick the negative because I want that my first term here to be positive so we're going to do that one how much is8 y 6id divided 8 y^2 remember ID negative is a 8/ 8 is cool we can do it that way how about y 6 / y^2 your first term should be y 4 for how many people is it y 4 did you get that can you double check it add these because they multiplying common bases that's why the 68 y^2 hey yeah we got exactly the same thing next up should This Be A Plus or or minus why pull out so remember we divide term by term 16 y 4 / 8 y^2 positiveid negative is a negative that's why we have the minus how much is 16 / 8 how much is y 4 / y^ 2 when we divide common bases we subtract the exponents if you have trouble doing that in your head that is not a problem write it out do not just say man Mr Leonard isn't in his head so I have to that's not true um if you have trouble with this all you got to do is just take this 16 y 4 -8 y^2 and then get your -2 y^2 that's where we're getting that okay so please do not make the mistake of thinking all this stuff has to be done your head it's not true next up uh we're going to do 8 y^2 divided by how much is a negative divided by NE 8 y 8 y^2 are these relatively prime they have to be because that's a one there's nothing else is going to go into that so you're automatically done you would double check that by Distributing but I'm sure you've already done that next one oh let's find our GCF here we how many terms do we got what goes in what number goes into both of those terms what do you think five write the five cool uh what variable goes into both of those terms that's your GCF that means that we're going to create some parentheses because we're factoring them let's divide 5 x to 4th divided by five how much is five divid by five one you're gonna write the one no you can it's there how much is X 4th / X remember this is X the 1st right so when I divide by 5x to the 1st that's one that's x to the 3 and that way our exponents are going to add half the 5 or two distribute okay lastly am I going to have a plus or a minus here can someone out there explain to me why in this case I factored out the negative but in this case I didn't why because you would create a negative on the first uh variable if you factored out a negative yeah that's exactly right if I were to do that and to make this divide by a negative my first term would be negative I don't want that this is already in order that's perfect we we like that so factor out a positive you're good to go uh now 20x divided by let's do this if you just want to write it right underneath second term divided by GCF 20 ID 5 how about x / X well that would be 4 * 1 4 * 1 is four we're just going to leave it as that four show hands if you got both those correct did you are there any questions at all before we do the next one did you do the next one yeah it's a little weird right there's so much going on is it super hard no not if you understand the GCF it's all the same same you just got to be careful so let's look at the GCF I'm going to write this one out that way uh that way I don't lose it because man this is it's a little bit more complicated so first thing let's look for uh oh what do we do first GC I know it's GC numbers let's do numbers we get eight 20 got 12 GCF true false is two noal four true okay eight no why not eight it won't go in 12 or or 20 but yeah for sure so four is going to be our GCF as far as the numbers are concerned now do we have to do all the variables at once no can if you're really good at it that's cool uh let's do the X's first so I X2 I get X 3r I get X what's my GCF for my x's that's the largest power of X that all three of these terms have in common how about the Y's what about the Y's y yeah I can't use y the 4th these guys don't have enough to go to lunch so y the thir that works here that works here that works there were you able to find your GCF of 4X y 3r all we going to do is divide it so our GCF goes in front we create some parentheses we have one two three terms we're going to end up with one two three terms um right now it's a this type of a problem because we have all these variables flying around we really don't need it specifically in or order as far as far as X go because it's not a right now it's not that type of polinomial that we do that with so if this is the case if first term is positive just factor out a positive that's that's basically the idea you guys with me on that one so we're going to go ahead and do that what might be a good idea here should if you if you struggle with this would you try to do this in your head if that's if you struggle with it you probably write it out or you can even do this do the 4X y 3 if that helps you can put it all three spots that's cool too how much is 8 ID 4 how much is x^2 / X how much is y 4th / y 3 cool we subtract exponents every single time when we're dividing by Common bases next up plus or minus remember that we're dividing by always the same GCF you cannot change that 20 ID 4 x 3r / X and lastly Y3 Y3 do I have to write anything here cool I'm done 4 x y 3 12 / 4 so I'm going to have that plus 3 well the rest of it kind of nice x / x 1 y 3id y 3 cool we just have that three and you can certainly double check that by distribution what I want to know is have I explained up to this point how to use the GF for factoring well enough for you raise your hand if I if I have feel okay about it okay now y'all love fractions right oh fractions are horrible because they make things more difficult so one thing that happens is when you're factoring a lot of times it's really nice to factor out the fractions because that makes the rest of your problem inside your parentheses easier so when I'm giv you a really easy example to illustrate this um it is harder if these are different numbers because you have to factor basically um a little bit differently uh and I'll show you one of those a little bit later but for right now if you have the same denominator like in this case nine look up here are there any numbers which go into five and one and two one besides one so our GCF here you said one right one goes all one factor out that that common denominator do that it'll make the inside of your parentheses much much nicer looking now the VAR variable should be easy for us real quick what's the GCF for variables if we factor that the 1 nth x to the 3r I'm going to do it really fast for you guys because I believe that you have this kind of down if I factor out the 1 nth yeah you could do 5 9ths divid by 1 nth it looks kind of awful uh to be honest with you but remember that all that's really going to happen in this case is you're going to multiply by the reciprocal of that fraction your nines are going to be gone so when you factor out the one night it's really the nine that's disappearing you're going to have the 5 x 5th over X 3r remember you subtract when you divide common bases that's X2 we're not taking any signs away we're going to have the same signs 1 nth when I factor out 1 nth well that's one we'd have X because x 4 over X 3r gives us X and lastly that's a minus this is a minus 2 9ths / 1 9th all it's going to do is remove the 9th part of that we're going to have two and then we have x 3r / X 3r that's going to be oh man I lost half of you did that could gracious uh x 3r / X 3r is is it's one hopefully not zero 1 * 2 we got two we okay with those ones the two if I'm basically what I'm doing I'm removing the one nth from each I'm I'm getting rid of the fraction so if I remove the one nth take take away this idea of having the fraction one nth I get five take away the idea of having the fraction one nth and dividing that out I get one take away the 1 nth I get two basically just the numerators that's all we have 512 hey 51 two that's the idea does that make sense to you I know it's a quick quick little trick but uh it's kind of nice question couldn't you um also multiply each one by nine and then it would cancel out the bottom denominator no you can't the only time you can ever ever ever multiply by a number that's not one is if you have an equation If This Were equal to zero if if it's an equation you can do almost anything you want want to as long as you do it to both sides oh wait wait wait a minute if I don't have an equation I don't have the idea of both sides does that make sense which mean you can't just multiply whatever you want to it' be awesome but you can't do that it's a good question you're in the right track but you're a little bit um thinking of equations rather than Expressions thanks for bringing that up though can we move on a little bit Yeah okay this next part is really really important because we're going to be using it a lot a lot a lot I like a lot uh a lot in the next few sections what it involves is something called factoring by grouping it's not it's not really hard but it can be a little confusing if you've never seen it before or if you've forgotten it uh because it looks a little weird U so I'm going to introduce spectum by grouping with this one example if you had this so factoring by grouping let me let me pretend for second that I couldn't combine those I know right now that 8x + 7 x is 15x we know that yes not 15x squ or something something like that but but 15x now let's say that what I want to do instead of combining these was I wanted to factor it could I factor it let's try this again could I factor it are you with me yeah what goes into but what's the GCF of of 8X and 7x so if I were to factor this I'd write the X here correct and then inside parentheses I would have 8 plus seven you all still with me now let me let me show you something about this how many X's do you have here how many X's you have here why did these two x's become 1 X because if you were to distribute It This One X would go to both terms wouldn't it creating these two x's again this idea right here if you if you understand that factor by grouping it's easy swear easy okay what happens is you have 2 X's here you have One X here because we're factoring and dividing both those terms by that one single Factor it makes it look like there's only one X but if you were to distribute it it would go to both those terms giving us back those two x's quick heading on you're okay with the idea all right that's how factor of grouping factor by grouping works so let me take this and run with it a little bit so again these 2 X's become 1 X because when we factor from both terms to undo that we would distribute to both terms you get those two x's back again now if we look down here yeah it looks a lot more complicated right because there's lots of stuff going on but I want you to recognize what we're doing you with me recognize what we're doing how many terms do you have here two two how many terms do you have here look you have two this is one term and this is one term yeah they're really big fat terms but notice how it's this and this that's separated by the plus do you get it what that means is that this is one term and this is one term what it also means is that someone's already gone ahead and done the work for you and actually factored it have have I lost you some you guys are zoning out still L you if you look at this that's multiplication yeah we all know that don't distribute this someone's already done the work that's a factor of seven that's a factor of X+ 3 that whole thing is considered a factor here here that's a factor of y That's a factor of x + 3 so when we're looking at this we're basically looking here and here and asking do these things have a factor in common a factor is a piece of the of a math puzzle piece of a term basically a piece of a term that's being multiplied by another number variable whatever another Factor do they have a factor in common yeah now check this out this x + 3 acts exactly the same as this x does The X went out front didn't it are you listening the X we did went out front were factoring creating even more parentheses so when I take this X and go okay look at that's a factor isn't it it's being multiplied that's a factor isn't it it's be multiplied I take that X I put it out front create parenthesis this now here is my factor that is in common it's the greatest common factor take that common factor and put out front now wait a minute does factoring destroy parentheses or create parentheses I already had one set I need to create another set factoring always creates parentheses so if I factor this now now check out what happened look look at are you guys seeing the similarity between these two problems I drew it up so hopefully you would if I factor out the X the only thing that's remaining is that eight do you get it if I factor out the X the only thing that's remaining is that seven if I factor out my x + 3 that's the factor they have in common what's remaining if I factor out divide out the X+ 3 seven is going to be there you with me plus plus hey plus plus plus plus if I factor out the X plus 3 the greatest common factor factor it out what's going to be why because I asked you little math humor I know I'm such a D anyway can you see that if you were to distribute this you get the same thing check it out x + 3 it's weird to to see but if you took x + 3 and distributed you'd have 7 * x + 3 take X+ 3 and distributed you'd have y * x + 3 show feel okay with that what I really want to get across do you understand why there are 2 X+ 3's here and Only One X+ 3 here that's what I want you to get do you understand why it's not squar is that x s No 2 x is 1 x 2 x + 3 is 1 x + 3 you're factoring it which means if you distribute it it's going to go to both terms you're going see it twice again if I've explained that well enough uh give me a quick head no if you understand that that concept we're going to practice a lot more trust me but I want to make sure that the the concept is right that way you understand this further you had a question Oh I thought I saw a hand go up sorry so if it was X squ it changed the problem if what was x s here yeah if this were x s and this were not x s you couldn't do it if this were X squ and this were X squ it would be no different that would be X squ all I'm trying to tell you is these two things have to be identical that's all you got to know if these are the same you're good to go if they're not the same you got problems you got problem man okay tell you what why don't we try a few more so yall get the hang of this in fact I'm going to have you do one right now just on your own I'm going to kind of lead you through this but you're going to do all the work here the first thing I want you to look for how many terms do I have two big terms yeah they're really big terms do those two terms have any factors remember factors have to be contained in parentheses uh do they have any factors in common yes what factors okay so write that Yus one in front did you get it how many y-1 should you have right now cuz you're factoring it's coming out of both next you're going to factoring always creates parentheses always create another set of parentheses did you do that already yes so you say where I go two terms one two anything in common yeah common factors is what we're looking for common factors have to be in parentheses or be multiplied so y - one is the common factor that's what we did before right wrot the GCF first now if I divide by y - one you didn't do it here if I divide by y - one well those would go away you would get simply X why well because you're factoring you're dividing out that y -1 it's going from both terms that's great uh 13 y -1 / y -1 what's going to happen-- 13 13 factoring always creates another set of parentheses and that's our answer could you check your work yes just distribute it that's the idea okay all right we're going to end there uh next time we're going to start on a little bit more advanced concepts I'm wait wait wait I'm going to give you ones that don't automatically look like this cuz this right now is really simplified okay we're start figuring out how to do that all right so we're going to continue our journey through Factory and what we were discovering was that uh when we have a GCF the greatest common factor it doesn't necessarily mean that it's just one number or one variable uh it could be an entire factor in parenthesis and that's what this factoring by grouping does for us it says you know what even if you have something like this where this is one large term and this is another term separated by that minus even if you have these two terms we're still looking for factors and factors are really anything that's grouped together that's being multiplied by the remaining part of that term so for instance this right here that's a factor and that's a factor of these one two large terms and factoring by grouping I know we did this last time but it's kind of refresher factoring by grouping says we can Factor as long as those two things are exactly the same watch your foot please uh as long as those two terms are factors are exactly the same what that means is that we can remember what factor does right factoring is the opposite distribution so where distribution destroys parenthesis factoring creates parentheses a new set of parentheses what the factoring says is out of these two terms if they have a common factor the greatest common factor we write that common factor out front just like we normally would we create a new set of parentheses just like we normally would and then we find the remaining two terms by dividing remember that uh does factoring ever get rid of terms so if you start with two you're going to end with two inside of your parenthesis you might just have a new factor out front so here if I factor away my 2 m- N remember you can even show that if you you wanteded to divide by two sorry 2 N plus n if you want to divide by that those are gone what's remaining from that big term so from this term we get the 5x2 Y 4th quick head not if you're okay with that so far is that sign going to change when does that sign change if we do what so here it's still be minus we do the same thing here now wait a minute if when I factor this and I take this thing away so so to speak does that become a zero no now remember the smallest thing we can ever get is what yeah and because that's because we're dividing so if we divide this out that's what factoring is we're actually going to get a one we never eliminate terms you're always going to have the same number of terms inside the parentheses as what you started with we start with one two big terms we end with two terms inside that parentheses show up hands feel okay with so far good so just a little refresher to get us rolling today and now I'd like you to try one just on your own really similar idea um really similar idea but just with [Music] this okay go for it so I want you to I want you to verify how many terms are there first just look at the number of terms that you have remember terms are those big pieces separated by pluses and or minuses then what I want you to look for inside those terms do they share a common factor if they do factor that factor out so put that in front around pareses create a new set of parentheses and then divide to figure out what those new terms are going to be okay so let's go for how many terms do we have here two yeah two big terms are separated by that minus sign do we have a common factor what is the common factor that's right it's all contained in the parentheses that's what we would have to have happen otherwise we would have several terms here if those parentheses weren't there here factor factor in these two terms we write out our common factor first then we create a new set of parentheses because we're factoring and if we were to divide this away what happens when we divide out the a plus b what do we get left minus doesn't change what do we get here again yep we're dividing this so a plus b divid a plus b gives us our one and we're done show hands if if you got that one by yourself that's really good now we're going to make the jump uh right here this is basically been kind of done for you the first step has been done for you you don't know because we haven't talked about factoring by grouping in its whole but here someone's factored this so you have that that factor here and that factor here what I'm talking about are examples that look like like the following let's look at that example together now first thing I want you to notice is the the number of terms that we have how many terms do we have here four yeah there's four terms that's different than before right over here we had we had two terms here we got we got four terms now I don't want you to ever miss this ever first thing you do whenever you're factoring you always always look for a GCF if there is one always I don't care what you're doing you always look for that is that clear no matter what other factor I teach you about always look for the GCS so we're going to look for it does this have a GCF remember you got to consider all four of these terms does it have a GCF no no but that's what we check for first now if it had a GCF great you'd Factor it out if it doesn't we got to consider something else something else to do in this class if there's four terms and you're asked to factor practically the only thing we can do with that is try to factor by grouping so here's how you factor by grouping you organize these terms so that the first two terms have have a common factor and the last two terms have a common factor what that means is that sometimes I'll show you this right now if these things what was that a 2X 3 y that gave you yeah if the these things sure it was a 2X that was right if these things are out of order sometimes it won't make it difference and sometimes it will make a difference you might have to rearrange them in this case um it doesn't make a difference in some cases it will make a difference you would have to arrange them certain certain ways does that make sense to you so for us either way you do in this case we're going to group these so that the first two terms have a common factor and the last two terms have a common factor which way did you guys write it down this way or the last way [Music] 2 2us 3 y oh good okay good so they're grouped either way I just wanted to mention that sometimes you might have to look at at organizing them a certain pattern uh group the first first two terms and the last two terms basically what you're doing is you're trying to go from our four terms like this to our two large terms like these examples where you have a common factor that's the idea if it doesn't work the first time reorganize your terms and maybe it'll work a second time so for us uh look at our first two terms is there a greatest common factor here fact the X out now I'm going to start going a little quicker on our factoring because we talked so much about it okay so here yeah we factor out the X we' have y + 2 can you verify that for me yeah you sure okay plus sign plus sign it's not going to change now does these two terms have a common factor factor out the three we get y + 2 if it's factorable it will work out so that these two factors are exactly the same if that happens then you continue if it doesn't happen you try to reorganize your terms or it's not factorable quick head if you're okay with that so far what I want to know is are you okay getting the X factored out and getting the three factor out show fans if you okay very good now well wait a minute that looks exactly like what we did over here doesn't it this was it's exactly the same thing here's one big ter term another big term we had a common factor we're just going to divide out or factor that out what happens to my y + 2 what do I do with it yep so we have y + 2 I have parentheses what's what's the next step that I'm going to do just write X plus 3 factoring creates a new set of parentheses so create another set of parentheses if I factor away the y + I'm divid it I factor away the y + 2 I'm getting X if I factor away the y + 2 I'm getting plus and three I want to show if H feel okay with with that one now could you check your work yes if you distribute this here's your y * X or x * Y is the same thing here's your 3 y by foil here's your 2x right there here's your plus 6 it has to work so if you've done it right that's the idea behind factoring by grouping basically we've done this like mini like baby steps at a time I showed you how to factor a GCF I showed you how to factor with these awkward looking factors now we're just putting that idea together let's try a few more examples to really get this uh solidified and then then we're going to move on so for right now I want you to try one on your own I want you to look at at this here's the thought process for going through these the first thing you do you look at all your terms and you see if you have a GCF because if you do you need to factor that out first so Factor GCF if you have one for all the terms if you don't have a g UCF look at the number of terms that you have if you have four terms the only thing that we can really do here is Factor by grouping so you're grouping the first two factoring it grouping the last two factoring it and then you're looking for that common factor so go ahead and try to do that problem for if you would okay so uh first thing left side people does this have a GCF for all four terms what do you think no no yes yes what goes into all four of them oh no not for all four of them okay so that's the first thing we check for is for all four of them does it have a GCF because you got to do that first if it doesn't then we start looking at other other options does that make sense to you don't ever discount the GCF it's important to look for all four of them at once and then if it doesn't have one then we start breaking this up does the first two terms does the first two terms have a g there yes what is it if we factor out the X we divide it away we get y remember factoring cre parentheses divide out the X we get four good to go keep the plus here we have another factor that these two have in common we have a seven and if we factor that we get a y + 4 we could double check our work if we wanted to but that's the only way to factor these terms these two terms and the only way to factor these two terms quick head not if you got that far on your own at this point this is really nice now we've changed four terms into two big terms and then we continue we look for a common factor within those two terms so it's kind of doing like these two finding common factors and then another two terms finding common factors question there's two ways to do it if you do the XY + 7 y it does the same thing it does the same thing that's right sometimes that's going to work uh it does here and you can see that 2 * 3 six so no matter what way I have this organized 4 * 7 gives you 28 it's going to work for you sometimes it won't so if we continue to fact can we continue to factor yes how do you know because it has what if this was different than that would you be able to continue so that's a plus and that's a minus would you be able to do it that's a problem so here we factor out the y + 4 if we factor out the y + 4 from this large term what's remaining if we factor out the y + 4 from this large term what's remaining plus s good I like the how you said the plus seven right could you check your work I'm not going to do it but just foil that out it's got to be exactly the same as what you started with now let's try a couple more I want to really make this stick in your head here let's do uh this one question just my right side people over here question how many terms do I have and in those terms in all four terms is there a greatest common factor besides one for all those four terms anything go into all four of them besides one okay we check for it first but no not in this case so now that we have these four terms what's the next logical step if we don't have a GCF what should we be trying if we have four terms what should we try can you combine those are they like terms okay no so I'm trying to factor here if I don't have a greatest common factor and I have four terms the only thing we can do in this class is Factor by grouping so two terms GCF or some some very specific types of factor I'm going to tell you later three terms we'll talk about that in maybe about 10 minutes okay four terms GCF but then grouping it's all about Factor by grouping you don't if you don't have that GCF the only thing that we can do is Factor by grouping so I'm going to ask you one more time uh with four terms what's the only type of factoring that we do in this class grouping yep I want you to try grouping so we're going to group the first two and we're going to group the last two hey uh tell me something left side people what goes into both 3 A2 and 4 a what's the the GCF there hey you're right and when we factor that out 3 a 2 when I divide out 1 a what do I get remember I know I'm going faster right now you can always write this out don't be afraid to do this and say oh he's factoring out the a so 3 a 2 / a gives him 3 a that's how we're getting that do you remember that from last time so don't be free to write that out okay let's continue how about 4 a when I divide out that a what do I get B I'm going to get a plus between there quick head now if you're okay with with that one now this is a little this is a little weird though wait a minute over here we have 3 a plus 4 B 3 a + 4 what goes into both 3 a and 4 B one nothing nothing besides if you have two terms which are automatically relatively primed remember that word we're using it a little Loosely because we have variables here but they the idea is the same where these two terms have only the factor one when that's the case write down the one force it to factor by one I I promise it'll help you a little bit so if we write down the one and Factor this we still have 3 3 [Music] a plus 4 B I want you to verify that for me that that's still the same is that okay with you for some of you if you don't like the one that's fine if it does not Factor you can also just put parentheses around it just like that the reason that why we do the one is because sometimes people have people struggle with this if you if you notice this that problem is exactly the same as this problem except instead of a minus you get a plus do you guys see what I'm talking about so some people struggle that well why doesn't that just appear disappear why is that a why is that a one well you can always put a one up front so if you divide by this you automatically get one for some people this really helps to say the only thing that divides into these is one that that way when I factor out my greatest common factor here and here we don't forget that we create parentheses and we'll have an A and we'll have that one quick showe if you feel okay with that one do you need to put the one there no not really um you can if you want to I'm not going to get concern about it it's just kind of a little helpful hint so you don't forget that it doesn't go to nothing okay because a lot of people confuse factoring with I'm completely getting rid of it that's not what's happening we're dividing here make sense okay let's try just uh just one more together I'm going to give you a couple to do on your own and then uh now we'll be about done with this section okay you know what I'd like you guys to help me with this one I want you to tell me what I I know how to do this stuff I want to make sure you know how to do it so tell me what my first step should be when I factor no matter what all the time with factoring I'm always checking for what first very good and how many terms does this have four is there a GCF besides one that goes into all four of these terms okay but we check first never ignore that now we have four terms okay hey if there's no GCF and four terms and you can't combine them what do we use to factor it gr grouping that's right let's group them we group the first two we group the last two okay we're on roll now what do we do after I've grouped the first two and the last two what's my Next Step factor factor what each group okay so we're going to look at the first two terms and see if we can find a GCF just with those two terms and then last two terms and do the same exact thing so uh first two terms can you find me something that goes into just these two terms why CU I asked you I used that joke already didn't I yeah dang it oh I'm unoriginal now factoring creates parentheses what's remaining when I fact remember factoring just means divide okay we're just dividing 2 okay I like the 2x what's the rest of it 5 y very good if I factor out or divide one y from this y s i get one y left over quick head not if you're okay with it so far now the next part of this is really important this is where a lot of students make mistakes okay it's all about negatives um whenever you have a minus that is separating your two groupings Factor by grouping whenever you have a minus what this is going to do is is going to force you to factor out a negative you have no other choice so what that means you write down the minus CU it's a minus you can't change that what number goes into both 4X and minus or 10 4X 10 so because that's a minus it is going to force you to factor out not just two but -2 what that's going to do it's going to change your signs so do do you guys see what I'm talking about the -2 up there we always know that numbers go with the sign right in front of them so right here when we're factoring out two not really we're factoring out -2 because of that minus so when you do this when you find out your terms please don't just divide by two you need to divide by if that's a negative divide by -2 now have you understood that okay so let's do it what's uh what's -4x we always go with the sign in front what's -4x / -2 2 positive 2x isn't it okay what's -10 y / -2 do you know what M I bet you know I'll bet you know what most people are going to do here if they're going to make a mistake what's going to be there Mr Leonard's method don't work with crap man that's why I don't like this because it's it's going to be different if you mess up that negative can you continue with this problem right now that's a problem so it's really imperative that you understand that if there's a minus right here separating your two group groupings it is forcing you to factor out a negative all that you need to recognize is hey minus both these sign's going to change that'll be positive that sign will change plus just like that and what that's going to do is make this at least doable for you show P if you understand the the idea there it might take some practice to get used to the idea that minus means factor out a negative every time it's a plus don't worry about it just factor out the positive it's all good so now that we have this oh man we got one big term we got two big terms do you see see a common factor in there let's write our common factor first if we remove that common factor by division IE we're factoring if we remove it we have a y if we remove it we have a minus 2 and we're completely factored and that's nice what I'd like you to do right now is I'm I'm going to have you try two of them okay uh I'll be walking around if you need help now would be the time to ask so let's put one right here so we're looking at the total number of terms and figuring out if we have a GCF if we do awesome Factor it if we don't have that then we count the number of terms and if there's four of them we have to factor by grouping right now for for [Applause] MH think made I think so too what you guys can't figure that out come on it's going to be easier that's next right I mean I tested you you passed congratulations if you had done that problem then you wouldn't be able to go any further right and factoring by grouping would have failed so it's good that we see that but you try you try with four terms and if that happens you go well I just can't Factor it that can happen too okay so I'm assuming that since you're working on that one that you guys finished up this one is that true I'm going to go through it a little bit quickly since we should have this idea pretty much down to this point so we look for grest common factors X's here y's here nothing for all three of them so I'm going to factor by grouping because I have four terms I look here I see a factor of y That's the GCF for these two terms factor means create parentheses it means divide out that greatest common factor so we get our 2x we get our plus 3 y we never lose a term we always have the same number of terms here that minus is going to force me to factor out a negative something now the only number that goes into both these two terms is what one that's right I can put the one if I want to I don't have to put the one but what I do have to do is somehow make parentheses now if you're going to factor out1 basically I like putting the one because it reminds me that I'm going to have something down here not just nothing so if I factor out the ne 1 all that's going to do is change this sign to a POS 2X and change this sign to a plus 3 y you can verify that really quick if I take my NE 1 and multiply -2X 1 and multiply minus 3 y so we know we have it right now we look at a two large terms a common factor we write our common factor first we create some parentheses if we remove by division factoring if we Factor this out we have a y from that term if we Factor this out we have a minus one and that's the correct factorization did you guys get that one same thing happens over here it's just a a little bit different because we haven't seen this larger variable here no GCF but we do have a larger gra Factor than we normally have what goes into both those terms you got to factor out the X squ you have to do the greatest common factor not just X but you have to do the greatest common factor it's important for you guys to see that so if we factor out that X2 we actually get a 7 x by division X Cub divid X2 is X plus 5 X2 X2 is 5 if you don't do that it's going to look like these things are different okay you got to have the GCF now next one this is one of those cases where only one goes into 7x and five so at this point you can put plus and just put parentheses there 7x + 5 if you want to is only one goes into it if that looks fine to you do that uh if you think if you want to do like I do and say well I want the one there that way when I factor this it reminds me that I have an x s oh and I have a plus one and I lose any of those factors have I explained this well enough for yall to understand it okay cool I'm going to give you two cases where uh this might throw you for for a loop a little bit uh if you're not really careful I'll show you how to deal with those okay so first one okay real quick though real quick so how many terms four is there a GCF on these four terms no because some have X's some have y some have but nothing goes into all four of those terms so if we have four terms and I'm asking you to factor what do we need to use grp okay so check this out if we group this what goes in these two terms one what goes in these two terms do you understand that if I factor out one from both sets of those terms it's not going to do anything so yeah so this is this is one of those cases where one way doesn't work but another way might so if you run ac across this you go well wait a minute there's four terms nothing goes into those two terms besides one nothing goes into those two terms besides one this is one of those times where you'd want to try to reorganize it so usually usually it's just a matter of flipping those two inside terms sometimes it a little more complicated but usually it's just just remember this please remember this your terms have to go with the sign that's in front of them so if I'm going to Interchange this is it Min - x or plus X and then- 4 and then minus 12 Y and by interchanging them sometimes we match up factors better I didn't see that for what you did I'll get it do you see that nothing factors here mhm and nothing factors here right okay if we interchange the terms X goes here minus 4 goes here sometimes we're going to cheat not cheat use this to our advantage and then we get some some terms that do have common factors so let's look at it it again uh do these two terms have a common factor now and if we factor out the X we get 3 Y and we get plus one with me so far okay next up when I factor out the four when I factor out the four the first term is going to be one next term this one why is it going to be plus it's good this is important for you to see okay if that's a minus it means you're not just going to be factoring out four you have to factor out -44 id4 is POS 1 we think about it like -2 -2 y / -4 is positive or plus 3 y quick head if you're okay with it so far now next question do we have in these two large terms a common factor yes how wait a minute that's 3 y + 1 that's 1 + 3 y wait a minute addition is commutative which means that we can switch them around and make no difference so when you come across this yeah that's the same what if that was a minus would not be the same because those are not Comm subtractions not commuted so be careful on but yeah you can switch those around with no problem so we write our common factor first 3 y + 1 we're going to think about y 3 y + 1 and then we create a new set of parentheses and factor out for x- 4 so far so good on that one okay cool okay last one before we move on to a different technique here by the way the reason why if you have notic since taken us a long time to go over this the reason why we covered GCF to this extent and grouping to this extent is because you use GCF for all factor it's always going to be there and then we also use factoring by grouping in one of our techniques later on and so we're we're always kind of relying on these two two things question uh you mentioned like terms earlier will we ever encounter like terms while factoring you're it's going to be assumed that when you get something a factor it's already been combined like terms so like in our problems we're always combined like terms first and then we're Factor okay okay so yeah you look for that uh so when I say four terms it's kind of implied that they're four non- likee terms uh wow okay intimidating a little bit I don't know not if you know if you know what you're doing it's fine uh but you know if I gave this to you the first day in class said Factor it I go oh crap I don't know I don't know how to do that one well let's take a look at it let's take a look at this together what is the first thing that you check for all the time without fail when I ask you to Factor Anything GCF how many terms are there here four does this have a GCF with it please do that for first Factor the GCF first and here's listen I'm not going to show you the other way cuz frankly I don't have the time to show you a mistake okay but here's the mistake the mistake would be and I know you're all still working on it right now but just focus on this for a second okay a lot of people will skip this on a test and cause themselves more time you still get the right answer but a lot of people won't take the time to do it yeah here's what would happen if you factor by grouping right now what's going to happen is that within one of those factors you're still going to have a common factor which you need to take out later so take out the GCF now and it saves you time later it saves you a mistake potentially later make sense we're always doing the GCF first uh what is the greatest common factor in this example so if we Factor up five not only does it make all these terms smaller but allow it makes it so that later on we won't have to refactor again we don't want to do that so what I'm going to do is I'm going to make up some brackets really large brackets I'm going to factor out the five so basically just divide by five I want to know if you're okay with that so far okay next up is it true that this has no greatest common factor yes we've already done it so we can't have another greatest common factor we've already done that how many terms do we have still four four Factory never destroys terms so now we can go ahead into our faction by grouping so we look here and we look here just keep that five Flo loing out front don't don't lose it but you really don't have to work with it right now so if we Factor let's look at the first two is there anything that goes into both 3xy and 3 YZ we're looking for the GCF for just these two terms because we're grouping right now and if we factor out the 3 y remember divided by 3 y what's remaining for this term say what x X Plus X Plus yeah the 3 y we're factoring that way what is it plus perfect can you verify that for me real quick people that we uh we factored 3 y we get X Plus Z yes no yes we've already taken care of the five so we don't need to worry about that now next one is there anything that goes into both x z and minus z^ 2 Z oh okay so that minus that really drops down for us and it says if you have Z that goes into both these it's forcing you to factor in negative Z so so we're forced to factor a negative Z let's do that together uh everybody together what's going to remain here after we factor out a negative Z I like I said positive X that's great and somebody else who didn't speak just now if we have a negative Z and we're factoring out a minus Z or negative Z what goes here why does the sign change please because you pulled out that's right negative by negative is positive we're factoring out that negative did it work can you see that we made the right move here this is great we have two large terms we got a common factor we're going to keep the five but it doesn't really do much and put a bracket we're going to factor this x + z and put it first factoring always creates another set of parentheses if we Factor away the X Plus Z we get 3 y if we factor with the X Plus Z we get minus Z I need to show F you feel okay with with that one okay what happens if you don't do the G GCF is down here depending on which order these are in one of these two large factors like here or here will still have a common factor and you would have to factor that out anyway does that make sense one or maybe both in some cases but taking care of the GCF first make sure that when you get down to this point you're actually done and you don't have to keep rechecking so we always check GCF first it makes our lives easier so F if I explain factoring by grouping well enough for you that actually takes care of section .1 I know it's been a long ordeal but we're going to jump right into section 6.2 right now and talk about how to factor when there's not necessarily GCF but also when there's not four terms so we're going to talk about how to factor when we have trinomials basically try meaning three nomial meaning term so when we have three term pols