hello welcome back to algebra we're going to cover the incredibly crucial topic called factoring the greatest common factor from a polynomial this is part one of two will introduce the concept we'll work some important problems here to get your skills built and then we'll work some more complicated problems in the next lesson so I want to define what factoring is for you tell you why it's important and then we're going to kind of go from there right the good news is is very very simple to understand the bad news is factoring in algebra gives most students problems in the beginning it's one of the very first things that you encounter in algebra that I can't even describe it without showing your problem but it just gives students it just it makes life hard for students sometimes because up until now we've solved equations we've learned how to multiply things we've learned how to add things all of that's kind of cookie cutter with factoring you have to think a little bit more creatively sometimes you have to kind of like try a few things and be willing to fail a little bit to get to the right answer and I said fail is the really the wrong word but you have to be willing to try a few different approaches to get the right answer so what is factoring factoring and you're gonna get a definition in your book that's fine I'm gonna show you my definition which i think is a little easier to understand factoring is when you're trying to figure out what things will multiply together to give you whatever whatever you problem you have another way to think of it a better and a simpler easier way to think of it is factoring is a distribution in Reverse right let me say that again distribution and reversed let me say that a third time factoring is distribution in Reverse so let it sink in if you can remember that it'll be very simple remember what distribution is it's when you have a term in parentheses and you have something outside the parentheses and then we take and multiply that thing inside and we distribute it to each of the terms inside the parentheses that's what we call distribution now with factoring we're going to take the answers that we got to all those problems and we're going to go backwards and we're gonna start with the answer and we're gonna reconstruct the parentheses and the thing that sits out in front to go backwards from the answer to get to kind of reverse distribution that's what we call factoring all right so let's give a couple of examples I'm not going to write this definition of factoring down because it's not a really mathematical definition but ultimately factoring is distribution in Reverse and that's what I want you to remember so let's give an example really here let's take a term and algebra that might look like this X plus 2 now you look at X plus 2 X is a variable and 2 is a number they're completely different X is like apples and 2 is like oranges you can't add them together you can't do any multiplication I mean everything is different between these two things because this can represent anything and this is just a fixed number so because there's absolutely nothing in common between this and this there's no commonality between these two things at all you cannot factor it you can't factor this expression it's impossible to factor Y because there's no commonality ultimately factoring is all gonna be about looking at every little term and figuring out what is common to everything what is common to everything I'll say that third time what is common to every term that's what we're gonna be looking for but because there's nothing common here we can't do anything so we just write it down I wrote that down to show you that sometimes you can't factor things but let's take a very simple change to this problem let's take 2 X plus 2 now let me ask you something is there anything common between these terms well the first thing you noticed is there's an X here but there's no X there so X obviously is not common to either one of those terms but obviously the big elephant here is that you have a 2 which is also in this term multiplied times X and also by itself now agree they're not exactly the same because there's no X here but the numbers are exactly the same so what we want to do is we want to identify what is common to both of these things and we want to almost like we can reach into both of those terms and we want to yank it out and we want to we want to figure out what is common to both of those things everything joined by a plus sign we want to figure out what's common in this case it's 2 and we're gonna reach in with some hands grab the twos and we're gonna literally pull them out that's what we call it pulling out the twos pulling out the factors we'll talk about a little more about factors in a second but 2 is a factor of both of those terms guys why it's called fact we grab the tooth we yank them out so what happens when we do that is we take those twos and we pull them out but we can only grab one two from here and one two from here so we pull them out as a common thing and we have to open up a parenthesis and we have to figure out what is left over here such that when I take this two and I multiply it by whatever is left over it gives me what I started with because they're equal these have to be equal so two times what will give me 2x 2 times X will give me 2x so I have to put an X here and then I know that there's going to be a plus sign because there's a plus sign here and then I know if I distribute this 2 times X I'm going to get this 2x but 2 times what here will give me the 2 here 2 times what is 2 you know we'll know that 2 times 1 is 2 so this is what we call the factored form of this and now you can see very clearly why I told you the real definition of factoring that's easy to understand is its distribution in Reverse if I give you this problem and I say simplify this expression you know how to do that we distribute the 2 into here giving 2x and then into here giving 2 times 1 is 2 which is exactly this 2x plus 2 but now we're going backwards I'm giving you the answer and I'm saying what can you pull out that's common to both of those things yank it out and then figure out what must go in here to make them equal so literally what you do is you figure out what we call the greatest the biggest factor here so let me go and write a few things down which kind of definitions here so let me draw a little arrow here and a little arrow here and tell you that these things are called the greatest common factor now almost everywhere you'll see this representatives G see a greatest common factor of each term I wouldn't write that time right there now as well of each term why is it called the greatest common factor think about what those words say greatest means the largest right common means common to both of the things right and factor is what we said factor is something that's multiplied by something else to give you something so two is the biggest thing that's common to both terms that obviously because it's common to both terms it's going to be multiplied by something to give me my original thing that's the factor part of the factor means kind of like multiple multiple multiple it ativ term right then anyway it's the biggest in this case that's just a number but it's the biggest number common to both terms that's what it means greatest common factor of each term all right and factors in general are numbers that are multiplied together to give us something so we can just take that to pull it out figure out what goes in the middle this is what's called the factored form so if you work on a test or something and the question was factored this expression then you would just circle that that's the final answer the good news about factoring is that you always know if you're right because all you have to do is take your answer and multiply it out again and you should get what you started with otherwise you've done something wrong so you can see right away it's 2 X plus 2 boom you get the right answer now let's just move through this you're gonna see quickly quickly quickly how easy this is once we do a lot more examples what if we have change it a little bit 6x plus 3 right so what we're going to do is we're gonna look at each of these terms and we're gonna figure out what is the largest common thing the greatest common factor of both of these terms well X is not even present in both of the terms so X cannot be a common factor so it has to do something with the numbers but then you say well wait a minute these numbers are different 6 & 3 are different they're not 6 & 3 neither one of them is common to both right well let's talk about it a little bit more and see if I wanted to figure out what this term here 6x really is what do I have to multiply together to give me 6x well we say we're gonna list the factors the factors of 6 times X is 6 times X notice I drew this little tree here this means that it's pointing up here to the 6 X so I'm gonna write down here what multiplied together must give me this and it's obvious because 6 times X gives me 6x however the factors here we can say are let me take away the multiplication the factors are 6 and X however 6 can also be lit up into factors of its own right because there are other things that can be multiplied together to give me six right other than I'm talking about factors other than six and six times one is six okay I'm talking about other factors right there's other things that can be multiplied together to give me six what about two times three that can also be multiplied to give me six so you see that when I really look at this first term the factors that can be multiplied together to give me 6x really are everything at the bottom of this tree two times three which gives me six times X that is really that what we call the factor tree right two times three times X will give me 6x right so these are the factors that matter here right now four three the factors of three I'll just write it here the factors it's just gonna be one and three one times three right gives me three I can't pick anything else that multiply together the other two give me three so remember what the big picture here is I'm trying to figure out what is the largest thing common to this term and two this term but you see there's a three that's a factor here and a three that's a factor here so in fact I am able even though it doesn't look like there's anything common to these things it's telling me that the biggest thing common to both of these things is actually the number three it's the greatest common factor because three is present in both of these terms in terms of the factors that make up what is multiplied together to give me these terms that's what a factor is it's just whatever is multiplied together to give me this stuff three is the biggest thing common to both of these things then I open up a parenthesis and I have to figure out what goes on the inside then you do distribution in reverse three times something will give me 6x well 3 times 2 is 6 and the X has to be there to give me the 6x the plus sign is here so I have to join it with a plus sign 3 times what gives me 3 it's gonna give me 2x plus 1 and then you can always check your answer because if I cover this up 3 times 2x is 6x 3 times 1 is 3 that's exactly what I started with this is what we call the factored form of this so we're starting with simple problems but every problem of factoring will look like this you just look at the terms figure out what is the largest thing common to every one of those terms it has to be common to all of them in order to pull it out once you pull it out then you wrap your parentheses to write your parentheses down and figure out what must be present in the middle to then make the multiplication correct right and it's hard to explain why this is useful now I know you're looking at this like well why do I need to know this well I'm telling you that when we get a little bit farther into factoring we will be solving lots and lots of equations where you'll pretty much have to know how to factor in order to solve the equations up until now in algebra we've solve lots of equations it didn't require factoring but very very soon we'll be solving what we call quadratic equations and those require factoring in order to even solve them at all so we have to get good at this which the good news is is very simple once you get the hang of it now let's move over to the other board and write a slightly more complicated problem what if we have x squared plus x times y and I say I want you to factor this and get tell me it's factored form now when students look at this well especially this they can very easily see I pull out the 2 no problem when they look at this they can very easily see okay I can pull out a 3 that's kind of common to both okay no problem if they're I explain what factors are but when students look at this oftentimes they're like oh my gosh I have no idea how to do this because I don't understand there's no numbers here other than the exponent but all you have to do is look at this term look at this term what is common obviously there is no y in this term at all so it can't be a factor right if it helps you you will get good at this without doing this down the road but if it helps you in the beginning just write the factors of each thing what multiplied together can give me x squared well it can only be x times X that's the only things that can multiply together be multiplied together to give me x squared so these are what we call the factors of x squared factors just means things that can be multiplied together to give me that that's that's what that word means and what are the factors of this thing well x times y it's pretty simple it's just x times y there's nothing else I can do simpler than that to write it down and there are no numbers involved so I don't have any of those so the factors of this are just x times X the factors of this or just x times y so then I look and say well what is the largest thing I can pull out from both these well I can pull out a single X from here and that would have been allow me to pull out a single X from there I cannot pull out an x squared from both of them because although there is an x squared present here there is only one X in this thing so remember when you're figuring out what you can pull out as a factor it has to be present in all terms of your expression so even though I could pull out an x squared from here I don't have an x squared here so I can't pull it out as a factor because it is a present as an x squared everywhere only one X can be pulled out as a greatest common factor because it has to be common to all terms so I can only pull out a single X and then I just realize now I've got my factor and now I'm gonna figure out what needs to go in the middle here to make it true x times something will give me x squared that's something is X there's a plus sign here so I'll join it with a plus sign x times something gives me XY that has to be Y and then you double-check yourself x times X gives you x squared x times y gives you X Y this is what we call the factored form of that expression all right now as we crank through some future problems I'm going to take the training wheels off I'm gonna stop writing down the factors underneath everything like I've written the factors down because as we go forward you will get very good at looking at terms and figuring out what the factors are without writing these trees down everywhere but in the beginning it is really helpful so that's why I did it but ultimately if you get stuck you can always go back and write down these little trees to help you figure it out so let's increase the complexity slightly what if I had 10 x squared plus 5x squared Y and I want you to write the factored form of this so what would be common as far as numbers to both of these expressions right well 5 times 1 is 5 5 times 2 is 10 that's the biggest thing that I can actually pull out because whatever you pull out has to then be multiplied by something to give you this right so I'm gonna pull out a 5 it's the biggest thing common to both of these guys and then I look at the variables there's an x squared here and there's also an x squared so I can also pull out an x squared here there's a Y but there's no wise at all here so he's not a factor of both of course the Y is a factor of this term but he's not a factor of all of that one so I won it all so I can't pull it out so that that is the largest greatest common factor I have and now I'm gonna try to see what goes in the middle 5x squared times something will give me 10 x squared well that something has to be 2 5 times 2 is 10 and the x squared is there then I join it with a plus sign and then I have 5x squared Y so 5x squared times something will give me this the only thing that makes sense is the letter Y then you check yourself 5x squared times this gives me 10x squared 5x squared times this gives me 5x squared Y and this is the final answer now the other thing I want you to do and get in the habit of is well you could also write it as I guess I should say 5x squared a lot of times we write like to write the variable first so you could you might see it written as y plus 2 in the middle there and does it's the same exact thing the other thing I want you to do is after you factor something out take a look at what's inside your parentheses and just to make sure that there's nothing else common inside of there that you might have forgotten or didn't notice the beginning make sure that you can't factor it any further and you see that you have a 2 and a y here and there's nothing common so you can't Factor anything else out let me give you an example of what I'm talking about why you want to do that let's let me do this problem again let me go here let me just say that I noticed that 5 was the greatest common factor so let's say that I accidentally noticed that or I correctly figured out that 5 was the greatest common factor of the numbers but I completely forgot or didn't notice that the X squares here were common factors also so if I do that then 5 times what gives me this it's going to give me 2x squared that will give me the 10 x squared and then over here 5 times what gives me this it'll be x squared Y so this is the answer but then I'm like I'm about ready to circle it and then I there's a double check at the very end I go in here and I say are there any other common factors to these terms and I noticed all I missed it there's an x squared here and there's an x squared here that I can pull out so don't erase and start over just write another equal sign and say well the 5 is here but then I notice I can pull this x squared out as well right and so then if you cover up the 5 x squared times what will give me this I need to have an X here x squared times what will give me this I need to have a y here right the 5 was coming down all along but I just basically forgot to factor something else out here so then you can see that this is exactly what we got from before so when you finally factor something always go into the parentheses and just make sure that you got everything to make sure there wasn't anything remaining that you missed the first time that was common to both of them all right so let's take a look at the next one what if we have we're gonna increase the complexity a little bit 3p cubed yeah 3p cubed times Q plus 15 P squared what is common to both of these as far as numbers the greatest common factor well you know that 3 times 5 is 15 so you know it's a factor of 15 and 3 is obviously here so it's a factor of itself so the largest factor that that can be divided into is another way to look at it both numbers is the number 3 so that's going to be the greatest common factor of the numbers the peas I have three peas here and PQ but I only have two here so because it has to be common I can only pull out two of these peas right and there's one Q here and there's none there so it cannot be a factor at all so the the thing I pull out must be present in both of them then I say this times what gives me this well I need one more P and then I also need a cube to give me this plus sign from here and then what times this times what gives me this I need a 5 obviously and the P squared is already going to come along for the ride so this is going to be the answer so it'll be 3p squared times this plus 5 and you just double-check yourself this times this will give me 3 P cubed times Q and this times this will give me 15 P squared and that'll be the final answer and you can always double check yourself is there anything common to these terms in the inside and there is not there's one here there's five here there's no Q's or P's over here so I can't pull anything else further out all right let's take a look and switch it up a little bit and what if we have something like this slightly larger numbers 49 X cubed plus 35 x squared Y and I tell you factor that now you have to kind of get good at looking at numbers and figuring out the factors of numbers right away but whenever you kind of think about your multiplication tables 49 and 35 immediately you should start thinking about your multiplication tables what can be divided into both of those numbers you know the biggest whole number that can be divided into those numbers that's going to be the biggest factor and the number 7 should pop in your mind because 7 times 7 is 49 and 7 times 5 is 35 if you can think of the larger factors than that and then use it but that's the biggest one I can think of off the top of my head right now so I'm going to go with it and if I get to the answer and I see that I can factor more out then I'll do it at the end but for now I know that 7 is the biggest thing that I can pull out so I'm gonna go and put a 7 here that's for the numbers now I look at the variables have three x's here and two X's here so I have to have commonality so I can only pull out x squared there's no y at all here so I can't pull it out now this times what will give you this well 7 times 7 is 49 so 7 has to go in there and I need another X I'm going to give me X cubed so I have to write an X in there you should double check yourself that this multiplies to give me this then I have a plus sign here 7 times 5 is 35 the x squared is taken care of from this so I don't have to do anything there but I do need a Y because I need to have a Y over here then you double-check yourself 7 times 7 is 49 x q x squared times X is X cubed 7 times 5 35 x squared will come from here and the Y will come from here so this multiplied out and gives me this and then I double-check 7 and 5 I can't think of any bigger factors that will divide into both so I can't pull anything else out x and y are completely different so this is the final factored form of that of that expression alright just a couple more just to get a little more practice literally I think we have two more here they're gonna get a slightly larger what if we have 25 X Y squared times Z plus 15y squared Z squared first thing you do is look at the numbers what is the largest number that you can think of that will divide into both of these those will be the biggest factors obviously 25 and 15 the biggest thing you can think of off the top of your head is 5 so put the 5 out there then you'll look at variables I have an X here but I don't have any X's here so that can't be a factor of both Y squared Y squared that's common so I can pull out of Y squared and then Z and Z squared so that's good I want to pull out of Z but I only have one here and I have 2 here so it has to be common I can only actually pull out one of them then I open my parentheses and I figure out this times something has to give me this so 5 times 5 is 25 and then I need an X to give me this I need an X here and then the Y squared is covered here and the Z is covered here so that's it for the first term I join it with a plus sign and then 5 times 3 is 15 and then I have Y squared here and then I have Z times I need to put another Z here to give me Z squared and then you double check yourself 5 times 5 is 25 X Y squared Z X Y squared Z that's good then 5 times 3 is 15 and then I have Y squared and then I have a Z squared multiplying through let me just double check myself and make sure I got the correct answer that is correct 5 y squared Z 5 X plus 3 Z that's correct I didn't say it wasn't tedious I just said that it that it's doable and it's easy to understand but you do have to be very careful and I do recommend that you put your fingers on your paper and just double check yourself if you just do this and move on to the next problem you will probably get some long answers last question 24 R squared s plus 36 R s squared times T want to factor this I want to look at commonality I look at this and say what can I divide into both and the first thing I think of the 6 right that's what I think of first is 6 6 times 4 is 24 6 times 6 is 36 I'm going to pull that out 6 and then I look at variables I have two R's here but only one here so I can only pull one out I have an S here and an S squared here so I only have one here but two here so I can only pull single s out and the T is present here but it's not present at all over here so I cannot pull it out at all so I'm gonna pull out 6 RS now this time something has to give me this 6 times 4 is 24 I need an R squared so that means another R here needs to be written down I need an S that's gonna come from the multiplication so I'm good there I'm gonna join with a plus sign and then 6 times something is gonna give me 36 6 times 6 is 36 I need a single R I already have a single arm but I need an S squared so I need another s here and I also need a T which means I need to add a T here so then you double check yourself 6 times 4 is 24 R squared s R squared s 6 times 6 is 36 r s squared T RS Square t so that's the final answer but then as a double check you look back and say well is there anything common to this well I have an R and s T the variables are good but 4 & 6 something doesn't look right there I can pull out something common to 4 & 6 the biggest thing I can divide into those things is the number 2 because 2 times 2 is 4 2 times 3 is 6 so I'm like oh man but don't erase your problem all you did was when you did this problem you didn't pull out the greatest common factor you pulled out a factor that was present in both but it wasn't the biggest one that you could have found so it doesn't mean you start over but you can do it a bunch of different ways I guess to make it clearer let's just say we're gonna focus on this answer if I gave you this problem how would you factor it well the biggest factor here is 2 so I'll just pull out a 2 what do I need to give me this I need a 2 R plus 2 times 3 is 6 St double-check and agree with me that this multiplied together gives you this 4 r 6 st right so then to write your final answer what you're really going to have here is you're gonna have the 6 RS from before but then you're going to be multiplying that times what you got here 2 times our sorry 2 R plus 3 s T so what you're going to have is 6 times 2 is 12 R s to R plus 3 s T so 12 R s to R plus 3 s T that's the correct answer that's the final answer what did we learn from this we learned that when we looked at the initial problem the first thing we thought of is the greatest common factor of the numbers was actually 6 but it turns out upon retro retrospection that the greatest common factor here is not 6 the greatest common factor is actually 12 because 12 times 2 is 24 you see it's here 12 times 3 is 36 you see it's here if he had thought just maybe for a second longer we might have realized that 12 was the greatest common factor and we would have written it down and just eliminated all these intermediate steps the reason I'm showing you this way is because very often you'll do this yourself I mean you just won't notice the biggest common factor but if you get the answer it doesn't mean you've done it wrong it means you have more to do factor out something else but then when you factor out and it comes into the front you need to be multiplying it by what's already there to give you the final answer at the end of the day whatever you get at the very end should be multiplied together to give you what you started with that is the concept of factoring we're going to be spending the next probably five or six lessons on factoring of different types it's because it's a really important topic and it's mostly important because we're gonna use factor to solve lots of different kinds of equations later on this factoring that we're doing is just the type of factoring we're looking at each term trying to find what's common to both terms trying to find the greatest common factor of both terms and then we pull that out and figure out what needs to go inside that parentheses to backwards multiply and that's why I open the lesson and told you that factoring is really distribution in Reverse or multiplication in reverse I want to take him want to factor it out so that we can see what we need to multiply by to give us what we started that's saying that out loud gets very confusing it's not so easy to understand but seeing it in problem form I think is a lot easier so solve all of these yourself get a pencil and paper make sure you can do everything from the simplest one all the way to the most complex one do them yourself follow me on to the next lesson we'll get some more practice with factoring expressions in algebra