okay so we're starting with an easy warm-up to the course hopefully this will be easy but really important um we're going to do a lot of factoring and we're going to start doing it soon it's going to kind of last throughout most of the course so it's something that you know isn't going to go away and it's important not only that you know we know how to do it but that so we're proficient at it meaning we can do it accurately we can do it well consistently and then also that we're efficient we can do it quickly and the reason for that is we do so much factoring we do bigger questions where factoring is a small part of it and if you can't do it quickly efficiently then you know it's just gonna take too long to finish all of the questions so any little tricks that come up any little uh things that you can do to save some time all of that is uh gonna be great it's gonna be stuff that you're gonna wanna think about and you know just be aware of and try using it as you practice and see if you can use those tips and tricks or whatever they are okay so this lesson is just about factoring some different kinds of polynomial expressions uh one of the first things and in fact in this course uh you will notice that i talk a lot about observation so when we get started on a question like i said we do questions that are big questions they take a long time to complete and when you first approach the question you're going to start to make some observations so you're going to say oh it looks like this oh it looks like this oh it looks like this i know this seems pretty cryptic right now because we haven't gotten into the course yet but you'll you'll see when we get there and so i talk a lot about observation what do you notice what jumps out at you like what's happening and that's gonna that's gonna dictate how you approach the question so in factoring it's you know some this is something we're gonna be very good at i think because you've been factoring since grade 10 uh but what you notice the first thing you're going to notice is what kind of factoring is it going to be so we've got a few different kinds of factoring questions that we're going to look at here and the first thing you have to determine is what kind of factoring is this how can i factor it which of these procedures does it fall under right which one of these categories does it fall under so that's the first thing that we're going to do and that's what this quick summary is all about so let's get into it um we're going to look at the different kinds of factoring how you can observe how you can notice which type it is and then when we do some actual examples we'll show the actual strategies the actual procedure of how to factor okay so one thing the first sort of thing that we should always be looking at is common factoring common factoring can have any number of terms okay two terms three terms four terms ten terms whatever uh always look first for a common factor there's several reasons for this generally speaking in this course we're not just going to give you a question like this and say factor there is we will actually learn some new factoring uh strategies and so for those few days the question will say factor the following fully which means you have to common factor if there is one but generally it's a skill that we're going to use to do other things and so common factoring first is going to have a lot of benefits um it's going to make the question easier it's you know going to make you be able to do it more quickly that kind of thing and in some cases there will be things we can do that not only make the factoring easier but make the whole question easier so you know there's lots of good reason to get in the habit of common factoring first so always look for a common factor first and look for factors common factor of numbers and variables okay and we're going to see that in some of the examples that we do in some cases you may not have it an expression might have a common factor and you don't even have to pull it out it won't matter so we're going to again something else in this course we're going to notice is there are less strict rules for what your answer necessarily has to look like that kind of thing uh but anyway that that'll make more sense when we get into that part of the course factor by grouping we're going to look at this a little bit later not really too much today um this is something that you do in grade 10 probably don't do much in grade 11 but there will be a few times in this course where it's going to come back and it's going to be useful to review factor by grouping these are the ones that we really want to get into today simple trinomial and complex trinomial and simple trinomial has three terms because it's a trinomial and the key to it that makes it different from a complex trinomial is the coefficient of the x squared term is one okay that's what makes it a simple trinomial compared to a complex trinomial notice complex trinomial three terms but the x squared coefficient is not one okay so it's got some number in front of it and the reason why you have to determine this and again this is going to be really easy once you get back in the habit of factoring this is a no-brainer it's not going to take you time to determine which one it isn't going to sit there thinking about it you're going to just know but it's a great but this is i really like um generally speaking this factoring if we step back and think about these little observations that we're making and those observations tell us what we're going to do next that's a really really neat skill and really useful and we're doing that more often than we're aware of and so that's why i still like to kind of talk about it and point it out so uh if you've got three terms you might be looking at a simple trinomial or you might be looking at a complex trinomial and we'll see some examples of those coming up and then the last couple difference of squares is a big one uh generally you've got two terms they're both perfect squares hopefully we know what that means hopefully we know our list of perfect square numbers and there's a minus sign between them or you know you could rearrange it to put the minus sign between them if it's not actually between them on the page uh we'll talk a bit about that as well um that's usually an easy one usually a straightforward one uh perfect last one perfect square trinomial you know you don't really need to know this one there are a few uh procedures that we do in mathematics that require the use of a perfect square trinomial complete the square is the one that we we start doing in grade 10 and that comes to mind um and you kind of need to know what a perfect square trinomial is for that but um otherwise you know if you notice that it's a perfect square trinomial it might make you able to solve the question a little bit faster again efficiency is good uh the other thing is um there will there's something we're gonna do pretty soon in this course actually where it noticing that will help us organize our solution in a way that's more helpful so you know you might notice it you might not in the end it probably won't matter too much and i'm not going to go into a lot of detail about this one right now but it is a trinomial so it's got three terms and then those three terms have to follow those two which you can't even see so i'll zoom out okay those two conditions of the pattern in the coefficients that kind of thing okay so when when it when it becomes more relevant later on we'll probably talk a bit more about this but for now we're not going to worry too much about it all right so let's dive into some examples here so as i said first thing we're going to do is we're going to make some observations what do we notice about this well i notice it has three terms so it's a trinomial but it doesn't look anything like the a simple trinomial or a complex trinomial it's got all these m's and n's and all these kind of crazy exponents and so you know it's neither of those and this is just a straightforward common factor question so what i do for a common factor question and we're never going to see another question like this after today you know we don't really work with expressions this is called a two variable um expression and we really work with single variable um algebra and single variable expressions so we're not going to see anything like this but it is good to review it gives us a good review of the the basic rules of common factoring i look at the coefficients first four eight and ten i think what number can i divide out of all of those the biggest number is two so that's the one i'm going to take out always take out the biggest then i'm going to look at the m terms and i'm looking at their exponents and again i want to take out the biggest that i can take out of all of them which turns out to be the one with the smallest exponent and so that in this case that's a one so i can br i could whoops pull out i don't want that bracket there pull out m to the one and we don't write the one and then same thing for the n's i've got a 4 a 2 and a 3. so i'm going to pull out the smallest and that turns out again to be in the middle term there which is n squared so i'm going to pull that out okay now i divide 4 divided by two is two m squared divided by m to the one so remember you're subtracting exponents when you're dividing powers of the same base so two minus one is one so it's m to the one n to the four divided by n squared is 4 minus 2 which is 2 so n squared and again the trick is and this is always good to think about with factoring it's something we're going to do when we start to learn new factoring rules is the idea is if i re-expand so if i multiply these two terms back together if i were to i'm not going to but if i were to i'd get back the original expression that i factored okay so you can always do a check negative 8 divided by 2 is negative 4. m divided by m is 1 it's gone n squared divided by n squared is 1. it's gone so that one's done 10 divided by 2 is 5. m cubed divided by m 3 minus 1 is 2 so m squared n cubed divided by n squared 3 minus 2 is 1 so n and i'm done and you know you might look again to see if there's another common factor there but if you did it right the first time there won't be and you should be good if there is then you should go back and fix that and make sure you factor it fully because that's going to be helpful i would imagine people might ask hey what about that um should i rewrite this and put that constant at the end you know again this is something that we're gonna we're gonna talk a fair bit about in this course because i'd like to talk about these things i don't have any set rules for i mean in some cases there are definitely rules for how you have to write your answers but for for the order in which you write an expression there's not really hard rules in mathematics you have to write it this way there are standards the way we tend to do it and that's in order to make our life easy it means other people can read your work more easily you're going to notice patterns more easily so like there's a real good reason to do it so that's how we're going to do it but in this case there's no reason to rewrite this and put the constant at the end like what what use is that we're not doing anything else with this expression we're not doing any more steps so i don't see the point if we were going further with it then that certainly is something you might do is rearrange it to write it in the correct order okay so that one's done common factor next one well hopefully we noticed this already hopefully this is familiar uh we've definitely got three terms and the x squared term it wouldn't matter if this was written out of order right if i if it was written like this negative 5 x plus x squared minus 14 that doesn't mean i'm looking at that first coefficient the order of the way an expression is written really matters this actually comes back to what i was just saying about patterns the reason why we write it in the same order is because it becomes easier for our brain to just look at the first one because that's what we care about um but you would always be looking at the coefficient of the x squared term whether it's written in the first position or the second and in this case it's a one so this is a simple trinomial okay and this is the pattern for this one is you're going to have two brackets and you're just going to have an x in front of both x times x is x squared right so that's why that works and now if you remember i'm looking for two numbers that multiply to the 4 negative 14 and add to the negative 5. and because they're multiplying to a negative one of them has to be positive and one of them has to be negative because when you multiply a positive and a negative you get a negative right so they have to have opposite signs which means they are the difference between them will be five okay so i'm going to be subtracting them to get the five so not many uh things multiply to 14 so this is clearly seven and two and now i have to figure out which one's positive which one's negative and i want the answer to be negative five so i need the seven to be negative negative seven plus two is negative five right and that's what i mean by the difference has to be five seven minus two is five and i know that they have to have different signs because the product was negative okay again observation like this is something we notice and then that helps us figure out what those numbers are going to be because we have to know that they're subtracting to 5 as opposed to adding to 5. all right and again you could uh check this by re-expanding x times x is x squared negative 7 times positive 2 is negative 14 and then positive 2x minus 7x is negative 5x right so you can always check and whenever you can check we're going to talk a lot about this in the course whenever you can check as long as you have time if it's a test and running at a time but as long as you have time it's always a good idea to check your answer all right so next one we looks like we have a trinomial again but this one's different because this one has a coefficient that's not 1 in front of the x squared term so it looks like oh it might be a complex trinomial but i bet people are already noticing that that 2 can be common factored out so we're going to common factor first and the reason why is that's going to make the next step way easier because now in this case this won't always happen but in this case now i got a simple trinomial and simple trinomials are a lot easier to factor this one's cool because when you have a 10 and a 24 there's a whole bunch of different combinations that might work whoops so 6 and 4 multiply to 24 and add to 10 right but 6 and 4 won't work in this case because it's the same thing i've got a negative here and that means they actually have to subtract to 10. so in this case my numbers are going to be 12 and 2. and again i want the 12 to be negative and the 2 to be positive so it's negative 12x plus 2x gives me the negative 10x and negative 12 times positive 2 is negative 24 and x times x is x squared okay common factor first uh be careful check your answers double check especially when there's some negatives in there even if we're good at this sometimes we're rushing too much we can make silly mistakes and get the wrong answer and again a lot of what we're going to do is not just factoring we're going to factor it then we're going to do all kinds of other stuff and if you factor wrong you may not be able to do all the other stuff and we have some questions in this course that are worth upwards of 20 marks just one question and if you make a mistake factoring in the beginning you're going to lose a lot of those marks just because you can't do the rest of it you can't show your work so it is really important that although i'm saying to be efficient and to be able to do this quickly we don't want to rush because we want to make sure that we're doing it accurately otherwise it could could cause some pretty major problems next one trinomial again and in this case though i've got another coefficient in front of the x squared that's not one but definitely not common factoring out so this is a complex trinomial and i should have done d and e in a different order maybe because i'm actually going to do a different kind of a trick to this one um and so if you need to go back and watch this again afterwards then come back and watch this part of the video again afterwards might be a good idea we have a particular way that we organize our solution to solve to factor sorry complex trinomials but there are times where i can cheat it's not really cheating but like i i can i can avoid that procedure and speed things up a little bit and those times are when either this number or this number are prime in fact in this case both are prime so that's even better uh if one or the other is prime i might be able to do this one more quickly and you might not feel ready yet but after you've practiced for a few days you'll probably you know a lot of you will probably be able to do this so what do i mean by prime i mean it doesn't have only two things multiply right five if you're doing two things multiply to five it has to be five and one so what i'm gonna do is i'm gonna put five and one right in there in the first plot i know that's what it has to be okay and the other great thing about this this is prime so i know these have to be a 2 here and a 1 here or a 1 here and a 2 here and i wouldn't actually i'm going to do this in a different color i wouldn't actually usually write this down okay i'm just writing it for the benefit of this lesson i usually do that in green when i'm doing lessons so i know it's one of the other and so in my head i'm able to and a lot of you are going to be able to do this just kind of do it in your head because you only have two options which one of those two things can i make add to nine and again in this case it's negative so they're going to have opposite signs so it looks to me like it's going to have to be a 2 here and a 1 here and the reason is i get 10x when i multiply those and i get 1x when i multiply that and so 10 minus 1 is 9. so that's how i know that i have them in the right spot and if i do this check if i have them in the wrong spot and i do this check i'm going to figure that out right and i just use a pencil erase fix it now i still have to do the signs do i want it to be 1 minus 10 or 10 minus 1 i want it to be 10 minus 1 because it's positive 9 so that means the 10 is positive so 5 times positive 2 is positive 10 and negative 1 times x gives me the negative x and 10 minus x is going to give me the positive 9x right so again always check because even i sometimes get the signs in the wrong spot and but i'm pretty used to checking my answer to make sure and um a quick check positive 10x minus x is positive 9x is always a good idea okay so in that case i kind of cheated a bit and i didn't actually write out the little system that we usually do for complex trinomial factoring uh and like i said because that's the first one we did that may have gone too fast for you come back and watch it again if you if you don't like it if you can't do it that way you don't have to do it that way for sure in terms of these kinds of factoring questions we don't actually expect you to show any work you don't have to so you could do it however you want um so if you do it this way great if you do it a different way that's okay too let's move on to the next one again this one trinomial it's got a coefficient in the front there and although i can factor something out there is a common factor of 12 and 6 that's not going to work for 17 17 is prime so this is a complex trinomial factor no common factor so let's get started and for this one now i know it's going to look like this for this one i'm going to do the whole procedure that's why i said maybe these are out of order but i'm going to do the whole procedure and hopefully you'll remember that now if you didn't take grade 10 or grade 11 at our school you may not have learned this strategy there are other strategies other techniques other procedures for factoring complex trinomials i don't care which one you use i think this one is the most efficient it seems strange at first and some people avoid trying to use it because it seems too hard but you know i think if it's too hard then i think this whole course is going to be too hard um uh it seems hard at first but with some practice everybody i've ever seen gets there and figures it out and it's a very efficient procedure for doing it doesn't require any extra work no extra steps it's all done in one step you do write something off to the side but and what but once you start doing it it also allows you to get familiar enough to start doing things like the cheating that i did over there so there's a lot of reasons to do it this way if you've never seen this before um go back and try to review your way set ask me next time you see me or send me an email and say you know i i couldn't follow that complex trinomial can you show me another way or can you can you go over it more slowly and i'll show you again but for most of you or maybe even all of you this will be familiar and this is i think the best way to do it so what do we do well i take this number the coefficient of the x squared and this number the constant and i write out there the pairs of numbers that multiply to those two numbers okay and i'm doing it off to the side i always do this kind of work in a different color so i do it off to the side so i i'm going to do it in order 1 and 12 2 and 6 and 3 and 4. those are all the pairs of numbers and you have to write it like this you can't write it like this okay two and six don't write it sideways right you have to write it the exact same way then you draw a line and the x squared the coefficient of the x squared has to be on the left and the constant has to be on the right okay those things all cannot be changed then i'm going to do the 6 on the other side of line one and six and two and three and then i have to flip one side or the other and i flip all of them and i'm gonna flip the right side because there are fewer pairs that's the only reason why i'm picking the right side and notice that i just flipped them both and now i have all the possibilities and i'm going to go through them to find my answer okay all right so that's the first step of writing this whole thing out it doesn't take too long right do it off to the side you don't have to do it i don't have to see it nothing like that okay how does this work i'm going to do one quick refresher you can go back and re-watch the video if you need to see it again and again or if it's if you're not getting it again ask okay what i do is i and this is easier to teach in in person it's a lot harder to do on the computer because you can't actually see me um but here's what we do i look at this pair and this pair so i always start with the first of each okay and what i do is i multiply crosswise so one times six is six twelve times one is and in this case because it's a positive 6 they're going to have the same sign and so they're going to add to 17. so then i ask does 12 and 6 add to 17 no it's 18 it's close but that doesn't matter so it whoops so it's not those two so then i'm gonna flip from this from from the one and the six to the two and the three and i'm just gonna go through every one one at a time and if none of them work then i'm gonna cancel that one and i'm gonna move on to this one and then do every one of these again and go through them one at a time and when you start to get good at this you can start to skip some of these steps and you know you're gonna you're gonna just know it's not that that's not those two so i'm not even gonna bother multiplying them you're gonna have a good guess of what it might be even this one actually i should say before i move on i'm looking at this and i'm thinking i bet this one is three and four and i bet this one is three and two and it might be one and six but i bet it's three and two and three and four and so you could put three and four in there and look at it and say can i get a two and a three in those spots and do it in your head and this this comes with practice you get better at it as you practice and some of us are going to be better at it than others some of you might not like doing it this way but you know you do get good at kind of guessing at what it probably is and you don't have to try every single one and you may not even have to write them out like this and again that really helps speed things up so i'll do this one 1 times 3 is three twelve times two is twenty four well again that's not going to add to seventeen so why am i even bothering right like why waste my time thinking about that one so i'm just gonna go ahead and i don't usually actually circle these you know you might you use your fingers when you're just practicing but i just kind of look and i i kind of decide well it's probably three and four and uh turns out i think it's gonna be here because three times three is nine and four times two is eight and 8 plus 9 is 17 right so that's going to be our answer now we read our answer across so my first bracket is going to be 3x and 2 and my second is going to be a cross on the bottom there four x and three i'm gonna kind of get rid of some of that stuff whoops i didn't want to get rid of that okay that's better now okay so i've got all that done i think i've got the right pairs now i have to put my signs in i know they're the same sign because they multiply to a positive and so they're both negative because they add to a negative and i now i'm going to do a check and as i said if you can check it's always a really good idea to check so 3x times 4x sure is 12x squared negative 2 times negative 3 sure is positive 6. and this is the bigger one negative 9x and negative 8x it add them together negative 17x okay so i got it and that's that procedure if you've never seen it before it seems a bit wonky the answer seems to come out of nowhere it seems magic you don't really understand why you're doing it but trust me it is the most efficient way of doing it and with some practice i've never met anybody who can't get there so uh you know if you haven't seen it i would strongly suggest reviewing this a few times and seeing if you can do it trying it the weird thing is you don't understand it so you don't even really know where to start and you don't know what to do to try and so it feels for a lot of people feel strange but you just have to you have to force yourself to try to write out the pairs of numbers and try to follow the steps and if you need a refresher ask me but you can use whatever you want you can use whatever method you want uh but it you know when you have to factor three four five times in one question if you're using a slow method it's going to be really hard to keep up so just keep that in mind that's why i like that method because it's the fastest and it gets us thinking in the right way to be even faster like i did that one okay let's get the last few done another looks like another complex trinomial doesn't it but wait a minute what do you notice what do you observe yeah there's a common factor of a co of the coefficients and they're all even so i can definitely factor out two but i'm like hm i wonder if i could factor out one or four as well maybe i could factor out four and turns out you can so this is three x squared plus x minus fourteen so if you're not sure most of us probably don't know that 4 times 14 is 56 so do it on your calculator do it in your head and sure enough it does factor and i still have a common fact or a complex trinomial factor like you know it didn't turn into a simple obviously it won't always but look at those numbers now i got a prime out in front it's way easier so i know this is going to be 3x and i know this is going to be x okay and you can you can write out the pairs of numbers on the side and if you want to pause the video and try to finish this one that's fine go ahead and do that it's a good idea but i know that it's going to be 3x and 4x and again i'm going to guess that it's going to be 7 and 2 because i don't think it's going to be 14 and 1 because i don't really see how i'm ever going to get that to add up to 1 in the middle i'm pretty sure it's going to be 7 and 2 and i'm pretty sure the 7 is not going to go there because that's going to be 21 and again that's just not going to work so the 7 has to go there and the 1 has to go there and and oh sorry 2. that wouldn't work if there's one and so let's see if i can make that work well this is 6x and this is 7x and sure enough 7 minus 6 is 1. and i want it to be positive 1 so i want it to be a positive 7 and a negative 6 okay and then i'm going to get rid of that and there you have it 7x minus 6x is positive 1. 7 times negative 2 is negative 14. 3 times 1 is 3. right fantastic okay so that's the trinomial factoring couple more quick ones left this is not a trinomial so we're not doing a simple or complex what do i see i see two terms i see all perfect squares nine's a perfect square one's a perfect square a squared is a perfect square and i see a minus sign between them so this is a classic difference of squares how do we do it square root of the first plus the square root of the second square root of the first minus square root of the second square root of 9 is 3 square root of a squared is a square root of 1 is 1. i bet everybody remembers that a lot of people that's their favorite because it's so quick and easy what about this one okay this is our last one for the for today sure looks like a difference of squares but 8 is not a perfect square and 50 is not a perfect square it does have a minus sign between them but it doesn't matter if they're not perfect squares but i know people have already noticed there's a common factor i can pull out a 2 and now look at that they are perfect squares 4 and 25 are perfect squares so now i can factor this as a difference of squares and i'm done okay so like i said you get sometimes you have to see the common factor otherwise the question won't work last thing i'm going to do i'm going to talk about a couple more things oops almost about or wrong um so there is another way we could do this it's it's very minor okay what if i instead of factoring out a 2 i factored at a negative 2. well then i wanted would have ended up with 25 m squared minus 4. and so notice how the 4 became negative and the negative 25 became positive and so they just kind of traded places and that only happened because i took the negative out front why would you do that do you have to do this no definitely don't have to why would you do this well you're going to see later in the course there are times where you might want to do this it might make future steps in the question easier this is a more normal way of seeing things with the m with the variable in the front position and the constant at the end so maybe it's even just more familiar and this is just a good trick there's no rule that you have to but this is something that you should be able to follow and you should understand and start to think about when this might be useful and in this case now i can go ahead and factor it and it looks almost the same because 5m plus 2 and 2 plus 5m are actually the same thing right order doesn't matter but 5 m minus 2 and 2 minus 5 m are not the same thing but because of the negative out front the entire expressions are equal they're the same both of these answers are totally fine there's no one that's better than the other there may be situations where you want to have one instead of the other but for a question like this they're both totally fine okay so that's factoring in a nutshell like i said we're going to do a lot of factoring in this course so being very good at it is going to be important and there's a lot of practice do as much of it as you need in order to get caught up in order to get warmed up before we actually start the course and you know every little bit is going to help trust me you're going to want to be good at factoring once we get there like i said there's going to be a lot of it all right that's it good luck