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factor so let’s say we have x to the. third plus eight and i tell you i want you to factor it now you’re very used to trinomials and factoring those but what about something that looks this a lot of . Mystudents would say can’t factor it and i get that it looks very different from something you’ve seen before but if those of you watching this have learned about the sum of cubes and the difference of cubes this is gonna look familiar now remember mathematics is all about patterns right recognizing .
Patternsseeing if there’s a pattern and i’m not . Gonnalie the first time i look at this it looks can’t do that can’t i don’t know what i’m gonna do i don’t know what i’m gonna do i’m used to something you know x squared minus 4 and then i know that’s gonna be what x plus 2 times x minus . 2but this is a plus now and and it’s cubed and i have no idea what to do so i completely. .
Understandthat so what am i telling you is look to see if . Younotice anything about it okay what do we notice about this binomial here i see hmm i see two. cubes i do i see x to the third plus two. to the third okay so we can look for patterns what do you see this is raised to this four x is raised to the third and then eight another way to write that is 2 to the third okay so then right now. what you might be saying is oh my gosh i . Rememberin . Classmy teacher taught me . Aboutspecial patterns i .
Canuse to factor the sum of cubes and the difference of cubes or a sum of cubes and a sum and a difference of cubes okay and this is clearly a sum of two cubes x cubed plus two cubed so let’s do a quick refresher on what that pattern is that we use for factoring okay a sum of .
Cubes.okay so let’s see right here i am going to kind of give it away a little bit you know what i’m gonna do here do a to. the third okay .
Plusi told . Myselfbefore. i’m gonna leave this undone i know it’s risky i want . Acolor on myself there plus b to the third okay and that’s. going to equal boost nope i don’t want this one i look one that’s going to equal a special thing okay um and so. it’s gonna . Bea binomial and a trinomial okay and the signs .
Go. Plusand then. minus plus and we’re gonna fill everything in there the reason i’m doing this number one is i really want to use colors because colors are really good for helping students just see patterns and it’s faster if i just am able to use you know the green and the blue to go quickly okay so .
It’sgoing to be a plus. b okay times the quantity a squared plus. a times b plus b squared okay so we can. factor a cubed plus b cubed as the. quantity a plus b times the quantity a squared minus a b plus b squared okay and if you click here you can see a where i talk a little bit more about that all right so now let’s see if we can apply that to this okay so i. already rewrote it which is a good first step i already rewrote it instead of x to the third plus eight i already rewrote it as x to the third .
Plustwo to the third okay so what we want to do right now okay it’s go first is we want to say that okay we want to say that a. equals in this case x okay and b equals. in this . Case2 okay so all we’re gonna do is we’ve identified our aarp remember what cubed plus what cubed well x .
Twocubed okay and now we can just apply this pattern all right so all we’re gonna do is we’re gonna say alright this isn’t too bad um i know it’s gonna . Bea binomial and then the trinomial and i know it goes plus minus plus okay .
Andthen wherever i. see an a i’m gonna put x and wherever i. see a b i’m gonna put two and i literally .
Justfill it in okay so it goes a let me see so that’s gonna be .
Xokay and then plus b which in this case. is two okay equals a squared so that’s. gonna be a is x so x squared minus a b. so the minus is already there a b so . That’sgonna be x and remember this is multiplication okay times two and then. plus b squared and remember b is two so that’s gonna be . 2squared okay and then. we can kind of clean this up a little bit alright so again i’m gonna go through with my black marker and put the parentheses and the signs in okay so it’s gonna be plus . Minusthat’s ok so. that’s gonna be x plus 2 there’s .
Notmuch we can do with that that’s just .