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9-5 AIM3 1.07 Getting Started

Added Sep 05, 2023 •

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We start looking at factoring techniques- including more advanced forms of factoring techniques you have met in previous classes.

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hello this last part of the chapter we're going to focus on dealing with various factoring techniques and I want to review them fairly quickly everything here that I'm talking about stuff you should have met before with possibly a couple of little I don't know tricks if you want to call them that but so let's get going all right remember how to factor a quadratic and cubic General reminders remember we love stuff in the Forum something times something equals 0 we can get a product equal to 0 we're in great shape cuz I mean to least one of those terms has to be zero that's one of the girls looking for can we rewrite it as a second thing is I just can't emphasize this enough and I hope it's been completely locked in your brain first rule what is the first rule of factoring that you will do this I miss they always got to be tricky with always never statements but I'm going to go for it first rule of factoring all right hope something popped in your head it is is there a g Chef if there is you need to factor it out I can be something simple something like this if I had a B+ AC and I really want to think that is a * b + C and I think we'll see that in action on another page all right. Let's go through this fairly quickly so we call him manik manik because there's a one out here this this over here would be non Monica cuz our legal fishing isn't one answer this is a fairly straightforward I'm guessing that you've done this kind of thing where you said okay I need two numbers that have a certain product in a certain something I need them have a product of 8 and a sum of 6 when it's Monica we can look straight at those numbers there actual special case of the non-life I'll get to that in a sec and in a moment's reflection I'm sure you would come up with four and two such as you it could be factored as x + 2x + 4 + here we're happy cuz now we're back to our product equal zero and this is great that means into two of them drag it out a little bit I could then say well that means that x + 2 equal 0 or perhaps x + 4 equals 0 and then obviously we could solve and get x equals -2 x equals -4 and so that's fine now commercial with all this one-off show you a shortcut over here all right so what about this where it's non manik and there is no GCF I check 2 and 9 we're done no, x x x no x no g c all right so I'm going to Zoom not assume I'm sure you met this before and we're going to use what most folks around here call the British method thank you British I wish I'd known this in high school I didn't want it till after I was teaching part of that we did guess and check it was horrible okay in a sense it is very similar to The Monaco but Shiloh I hope I left myself and I promise we're going to the same packing we're going to look for two numbers that have a certain product and at the same time have a certain sum but it's not just going to be 10 and 9 like we didn't truly what the thing you have to do you didn't do it here when you first met these even a previous classes cuz you always had a one out here did you really need to take the lead coefficient and the constant you need to multiply them together that's the product were looking for we're looking for two numbers out of a Harkins 25 through there lots of candidates and then have to add up to 9 and so you might initially to all 10 and 2:20 but of course 10 + 2 is not 9 that sound you think about it I'm sure you come up with 5 and 4 so that means it can be factored and actually I got to carefully I forget what I've hidden from you guys well that's too much I'm going to pause here for a sec okay I'm back I forgot I want to keep didn't want to keep everything revealed at once okay so now as I'm sure you've met in the past and bring this over a little bit we're going to name it using our five and four we're going to rename the nine as 4x + 5x or 5x plus parts order it doesn't matter but here's our people off of messed up they get the they may get the numbers for especially if the signs are different they don't make sure they add up to the right value well I'm we know 4 + 5 is 9 okay I called Sister four step process the first step is the Pro the Second Step I called history right we're just rewriting the problem using our factors that we're going to do the next step which is the GCF and way it works is you make art problem into two groups allow lines going here first I'm going to look at the first pair then I'mma ask myself what's the GCF look at hope you would say to X then check a 2X * what gives me that and so phone then I'm going to look at the second pair I don't know Green and then I would say okay now here's a nice little trick whatever you found inside these princes is guaranteed to go in here if you found a pair that working you wrote it's going to happen and so you can put an x + 2 here then you simply she okay what time is x + 2 give me 5 x + 10 and hopefully obviously you would find far then we're going to use essentially the reverse of this if we have something times something but sometimes something and there's a common factor we can Factor it out well look there's an x + 2 in both of these pairs so if I factor out the x + 2 I would get a nextplus to there it is and what is this x + 2 Neva x 2 x there it is x + 2 x 2 x distributive property then the next exit 2 * 5 there's our + 5 and some now I have factored it it's expected that you understand British method and you can go through this very quickly now for the solutions we could of course now this case this obviously gives us -2 we need to be zero at this one perhaps used to doing this to x + 5/2 equals 022 equals -5 + x equals negative 5 hats and that's completely fine no problem whatsoever however I want to if you haven't seen for introduce you to a shortcut for something like this I'm it work here as well but this is so straightforward but the rule works so notice this Express Rider this particular factor is essentially of the form a X plus b something times X plus something to course you could be mine as we don't well if I asked you to solve for x you're going to bring the B term over she got a x equals the opposite of B and then you're going to / a c and x equals the opposite of your break knows this sign here opposite of that sign so no other words change the sign in your expression and doobie over a will look as if we do that here change the sign becomes - 5/2 so you if you use that you can go straight from this expression to the solution without bugging you this little bit of algebra here I'm happy either way alright so non monic quadratics Mana quadratics by factoring techniques but sometimes they're not going to for us and so we need to try something else so remember completing the square I remind you completing the square is because if we have something of this type x + a square pom pom pom we get this this is very special pattern Otis arms Central term hear the middle term is each of the things that's being squared with an X being squared + 18 squared we multiply them the extra day and then we double it if that pattern exists means we can work backwards and just a X plus a quan D'squared so could we do that here well let's try it alright what should we do first alright let me ask you did you think g c f the first thing you should do is divide by to make your life easier x squared + 8x + 5 nicer now you know star no factors of 5 that add up to 8 only factors are five more so I'm going to use the complete the square technique where I'm going to move this concert over by sbtrkt five so there we go now I remember this pattern this number and here is twice the product of two things being squared well X is being squared here so I need two x x x something will obviously that something is for so that tells me that I need for square here that's the thing that will make this pattern work 16 and then I simply add sick get over here of course balance my equation and because I've had it exactly the right thing that means I can now call this X Plus for quantity squared equals 11 and this is another those expressions were very happy with because after all it's a squared equals some number will then a must equal plus or come on plus or minus don't forget the plus or minus so many people do and a why lose points you don't have to plus or minus the square root of n so here we could say come on x 4 equals plus or minus and since 11 side a perfect square with a sleeve it is Route 11 so we get Solutions of -4 plus or minus the square root of a right now a little reminder of something what if our time by completing the square what if we had this and she'll notice there is not a common factor so they're a couple techniques they're really the same they're just variations so one thing you could do well first we might do is move the constant over like we normally do cells put a -6 year and then I'm going to factor out the three now it does leave me with a fraction here for it's okay I know how to take half of a fraction and so now what am I going to adhere well you know we'll go to the same process option 2 you can see it's very similar as well I really like it when there's an x squared here that's what I need x squared so when I just / 3 right now x squared plus we're near the screen and it's doing that x squared + 5/3 X + 2 equals 0 and now I can move this to over and do the completing the square process can see how very similar to this if I were to divide by 3 right now and have a -2 over here it's just a matter of how you want start off if you feel you need more practice you can come in and work in the yard check on resources online lots of options you a couple of things that you've seen before what do you call this right here well I sure would help you remember the difference of squares and it should be merely presentable for factorable as first things S2 in charge of first term plus second term first term mine in a second term can we recognize that can we ride is something squared minus something squared so for example suppose I had nine a squared - 16 well 98 squared after all is 3A quantity squared + 16 can be called 4 squared so yes this is of the difference of squares form and so I can then get first term plus second term first choice my second term and I'm done course I hope I checked for GCF first alright these guys help this is the difference of squares what's this right it could be the sum or difference of cubes and you would have discovered there's a form for this and somebody was kind up to discover this Force long time ago but I must introduce you a little mnemonic device with a memory trick to help you remember least I hope actually everything takes time to practice so how to remember what's what well it's always just a basic terms I've got an X in a while and then I got the basic terms squared then the two terms multiplied in the second room squared how do I know when it's plus or minus minus or Plus there's a neat little trick so you might wonder why I've got a bar soap here can you figure out what my loan demonic stands so if it's X cubed + y cubed if it's a plus and we have a plus here in a minus here if it's minus if it's the difference Cube we have a minus in our first and A+ in our second make it look a little bit nicer and here's an idea plus plus they're the same plus minus their opposite minus minus a minus plus they're opposite but it's always positive here same opposite always positive that's what we can use to use a choke for an apple here we have what's being cubed x- okay what's being cubed to so those are two terms so we all right first term and then staying so it must be minus second term first term squared opposite the two terms * 2X always positive and then the second term squared so two squared gives us four and boom it is factored into other ways of doing this you may have gone to other techniques feel free to share them in class tomorrow at 1 last to to share with you and has two ties back to our polynomial division but specifically when we're dividing by a linear Factor okay if you're dividing by a quadratic or cubed what have you this synthetic division won't work I also would remind you that this is always so the form x - A from this case actually wear x- positive to so you're not work so since we're subtracting positive we can do this this isn't it's a division basically it's a stripped-down long division and then ends in here I'm going to write the coefficients of all my terms I'm going to include any missing terms back put any zeros but I'm not missing any here 321 okay so I have a 125 - 7 and here's the process first thing you do is you simply carry down so just put the one then you're going to multiply can I multiply 2 * 1 well that's obviously 2 * 1 is obviously too then you simply add them to get for now don't you multiply again multiply chocolate making will know your multiply multiply 2 * 4 is 885 and 8 is 13 / 2 * 13 is 26 and Adam up we get 19 and we're done now what does that do for us well off and before the last term is low separator it helps to visually what this tells us is that if I were to do x - x cubed plus 2x squared plus 5x y 7 / x - 2 then this turns out to be the quotient and this last term is the remainder we can you send text to vision for this process and also some other stuff will come later down the road and so now we know and pack you could say that what we're saying is if we had x - 2 * what is this will that be one x squared plus 4x + 13 and then I add if the remainder 19 that should give me that's and check it out - 2 * 13 is -26 - 26 + 19 is -7. looks like maybe we did it right synthetic division is very convenient process but only if you're doing with linear back. I think I'm going to write that but you must know the process and it will help you all right so I certainly hope that if you're not familiar with since I've given you a practice it that you can quickly identify the difference of squares the sum and difference of cubes and you know the factoring templates you can quickly come up with it and you can do completing the square and non manak British method and monic quadratics these are things you just must know we're going to be building on them already that's it take care

Uploaded by Brian Macky.

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9-5 AIM3 1.07 Getting Started

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