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9-8 AIM3 1.09 Factoring Cubes

Added Sep 08, 2023 •

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hello as you can see the title today is factoring Cubes but I want to start off with a quick review of some of our quadratic factoring techniques so one should recognize the difference of squares something squared minus I think squared and we know the form so for example here does it conform to that pattern well I hope you recognize that in fact we could call this 5 a.m. to the third squared -11 squared and so since it does fit the pattern in which quickly say all right well then this must be 5 a.m. to the Third + 11 then five a to the third - 11 I really urge you to rewrite it like I've been here as something squared minus something score immensely helpful to have that staring you in the face right now this next one perfect square trinomials this is one that we can take care of two are sum and product approach but but sometimes there's a shortcut so it's nice to see it and that's right can I see the pattern we know that if we have one term plus a second squared or minus a second squared it fits this pattern and the pattern is you have something squared first-term something squared last term and the middle is twice those two things are squared so for example if you look at this example we can see obviously we have an X being squared and here we have a 5 being squared and sure doesn't take long to see that we in fact have two x x * 5 not writing and I was kind of normal form but since it has since it fits that then that tells us is just going to be x + 5 squared course you could have found a normal Pro what are the factors of 25 that I pretend you to find out they were five and five so that we have the sum and product approach we talked about that if you know the routes well then you're in great shape and if you know you can use them to help you recreate the problem. Okay. British method I would like you to pause the video while you factor this must make sure we're all good so please go for it and then come back when you okay well let's compare results now you know so I wrote -10x + 3x you could have written 3x - 10x it all would have worked out these two factors within reverse order that certainly doesn't matter. You just got to know these in addition to that and I let's move on to do this let us move on so we talked about some and difference of Cubes before our soap method we know if it's can be written as something cubed plus or minus something Cube first term plus or minus right depends on what pattern we're going same opposite always positive so first term second term first term squared product of the two terms and then the second terms great that's the pattern so with that in mind how to deal with out shoetime bill that I forgot what I had their oh well so 88 cubed becomes to a cubed + 3 cubed and so we would get first term she this is a mine 2 - second term then first term where'd you probably just take care of that by yourself you don't need to ride out necessarily and then opposite so now we have plus their product well that's a 6A and then plus the second thing squared and I actually that can be cleaned up I'm not going to bother to do that right now okay so that's factoring a summer difference of Cubes but let's look at another variation what would happen if you had x + y to the 0 or x + y to the first X Plus y squared well I certainly hope this 11 anything to the first is itself and x + y con D'squared you dealt with before that should be automatic and although it's early days of turned out to be a bit of a pattern Happening Here one which we're going to play in other times something interesting happens when you have x + y cubed x minus y cubed show me a I want to see if you can see any kind of pattern between these two comparing them let's first look at the coefficients they're the same one 33113 through another's one is All Paws which makes sense this is just positive because it has a negative and in our power we get some sign switching happening how case that's a difference and then take a look at the exponents our first obviously is going to be in executing our last term as a y q but I'm think that's a surprise but then the second term x squared Why did the first and then X the first y squared so it knows it goes X Cube x squared x the first and then well x 20 why is there a y the first y squared y cubed this Hatters one we're going to want to recognize cuz it allows us to take something that might look like this when I say oh my goodness how am I going to factor it well I'm not going to tell you precisely right tell you these rules apply so I would like to see if you can Factor this by identifying how it relates to this and then we can share results tomorrow okay so you might want to come home and do that before we go on to the next part all right now let's look at some variations again how we can solve problems or simplify should say we're not stopping we're just simplify so we recognize this back we just talked about it something plus something to the 3rd so if I use that pattern that we just talked about this is Xbox one so no first term que blah blah blah like so you may want to invest in your notes you can compare to what I'm about to run so we get this first term remember this is a difference right so first term and by the way I'll meet you back up for a sec I was circling here and we're talking about a difference of Cubes actually this is something cubed minus and just to make it abundantly clear we could think of this as one Cube so what's the difference of Cubes to have a difference of Cubes pattern first term different show me second term first term squared plus the product of the two terms while since this is the one that you gives us x + 11 * x + 1 and then plus the second term that can be write this down Plus charm square but because it's one of course it just looks like one and if we clean this up well this is just a x square this out and I feel there was your ex squared plus 2X + 1 and then x + 2 and then one more cleaning we end up with that so fine but let's look at a different way perhaps there's something called Z substitution so maybe this looks a bit messy to you what if we let Z is equal to x + 1 and so now I'm going to call it easy cubed minus 1 cubed all right I'll still the difference YouTube spinosus just a little bit cleaner so 1st - 2nd first squared plus the product of 1 * easy plus the second one squares that's 10K then now that we have that factored now we can bring back our identity here so instead of xenon call it x + 1 - 1 and then Z squared with course b x + 1 squared + 0 which is x + 1 + 1 it gets us wish we would have finished this we'd end up in the same place and you can see we get this line but it just is a little bit cleaner at the beginning it may make some of you feel them or comfort do some people like to do this kind of approach but either way is completely fine and speaking of either way is completely fine let's look at it another way what if we took our cubaknow I I will refer back this x + y cuando Cube so we're cubing a binomial what if we did that well if I got my bearings here so x + 1 cubed would be this right first term Cube then remember the pattern 1 3 3 1 2 3 I'm just looking here underline things just like this is this and that is just that so this compares to the pattern we discuss in our cubing a binomial then what happened so if that's all that's all done now how can we clean this thing up without X cubed is X cubed + we have a 3x squared and we just have 3x how old is to cancel each other out so I just get that I see if there's a common factor of x 2 x x x squared + 3x + 3 and if we remember there it is a different way of getting there we have different approaches depending on what technique you want to use you on become with multiple techniques and lastly speaking techniques they called the grouping technique which actually you've done but it's probably have been pointed out to you it's actually buried in the British method the grouping technique works like this show me a course to snow TCF take your terms and breaking into smaller groups and see if something convenient happens there sometimes for example in this case we might break it into the first pair and the second pair chauffeured nap I looked at the first parents that they have a common factor of x squared so the x squared times x minus 8 then look at the next parents at all to have a compact or just for the fact about the four x - 8 and then ye right common factor of x - 8 that means we get x - 8 x x squared + 4 sometimes in the group Technic it's not a pair and a pair at least we're dealing with for charms it could be a triplet in a Singleton and it may not be in the order so you might take a look for a minute see if you recognize anything here that looks familiar so if we were to rearrange this I hope but that looks familiar that's just a perfect square trinomial which means we can call this x + 5 quantity squared and then minus 5 Quebec and Highway and we don't and we're not quite done because now we have something squared minus something swear this is a difference of squares situation which means I could call this first term plus second term times first term minus second term and it might get the order my cat rearranged we might go xy5 there to put in, standard form but that is a factored form do you just have to be on the lookout for different possibilities a little bit of a puzzle by then. As you can see your I'm strongly encourage you to read the minds and action for you start this assignment review your notes read this then go ahead and get started I think you'll find some success already take care

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9-8 AIM3 1.09 Factoring Cubes

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