Three Polynomials You Must Know how to factor

all right in this video what I want to do is go over these three polinomial that I feel are the most essential for students to know and understand on how to factor because when we're First Learning factoring we start with quadratics right and we kind of get that routine down and then we go and move into polinomial and a lot of times it's a seamless transition but for these three examples it's usually for students not so seamless so let's go over each of these separately so you can hopefully understand the difference in these types of problems and what we can do to go ahead and Factor them so in this first example you can see this looks like a quadratic trinomial I have three terms but instead of it being raised to the second power it is being raised to the fourth power however there's a lot of similarities with factoring this problem than there is going to be when factoring quadratic trinomials so what I want to do is Let's A lot of times when students are struggling with a problem like this what I like to do as a teacher is just say all right you know what forget x to the fourth forget everything else let's just pretend with the numbers that we have it was a quadratic Tri okay so again let's just make up this problem maybe something that's a little bit more familiar to us now if I was going to go and Factor this again remember for quadratic trials I'm looking for what two numbers multiply to give me6 add to give me5 hopefully as your brain is turning you're like oh that's -6 and a positive 1 so I could Factor this as a product of two binomials to a x - 6 * a X+ + 1 and again you can multiply that back out to verify but that is going to be general form for factoring a quadratic like that so when we look at a trinomial that's not raised to the second power but raised to a higher power what we simply need to do is think about well I still need to achieve the6 and the5 right so having my trinomial separated into two binomials makes sense especially with keeping the6 and the positive one now one quick thing I really want to remind you of this the reason why I initially am going into to bomal time a binomial is because remember when you multiply this out your terms are going to add to give you that middle term right so we know x * X gives me the x^2 x * 1 is a positive x - 6 * X is going to be a - 6X and - 6 * 1 is A-6 these two terms combine right now again my middle term is what these two middle terms combine to so what I want you to think about is look at what my middle term is yeah I'm going to get the neg5 right but I need it to be in x s or think about it this way I need the first two terms x * X multip give me an x s right now I need in this case to get it to an X to 4th so what should I choose now x * x what is going to give me I need this to be X to 4th now sometimes you might think well what about if you did x to the 4th and and or x to the 3r time x right that would actually add because you add the powers to X 4th however you wouldn't have your middle terms combined because again these terms need to be exactly the same because we need these middle terms this 6X and this one to go ahead and combine so these Powers have to be exactly the same so what two powers that are exactly the same are going to give me X to 4th hopefully you recognize it here and it's just going to be a X2 * an X2 and ladies and gentlemen that's it right now again let's multiply this back out X2 * x^2 is x 4 x * POS 1 is going to be an x -6 * x^2 is going to be a -6 x^2 and -6 * 1 is going to be a -6 you can see these are going to combine to give you back my original middle term Ah that's supposed to be an X squ sorry about that so all I simply when you're factoring something like this when you see it as a trinomial all you simply want to do is Factor it just like how you know how to do as a product of two binomials but then raise your powers to be able to match your middle term as well as to make sure that they're going to multiply to give you that first term obviously if that is not possible then this Factor technique would not work in that case all right now let's going into a polinomial with four terms now when whenever I'm teaching polinomial with four terms or at least factoring polinomial in general whenever I see a polinomial with four terms I just tell students think grouping right remember for quadratics when we see a trinomial we're thinking about doing you know that factoring technique and when we see two terms we're either thinking factor out the GCF or a lot of times I always say look for difference of two squares two terms look for difference of two squares or the GCF when we see four terms I immediately tell students look for grouping now again what grouping is going to do is just is going to group the first two terms and the last two terms however we have to make sure when we're grouping them that uh we're not changing the signs right and we're not changing it to a multiplication problem so what I'm going to do is I'm actually going to rewrite this just real quick because don't you guys agree minus is the same thing like plus and negative okay now the reason why I did that is because I want to group the terms but I still want to have them separated if you just put parentheses around here and you don't include that plus sign then you're going to turn into this into a multiplication problem and that's not what we're trying to do so when we're looking for four terms group the first two terms and the last two terms if you already have a subtraction just make sure you change it to a plus and negative so therefore you can group it correctly now the next thing we want to do is just look into factoring out the GCF of each of these terms so I say all right what does a 6x^2 and a 4X have in common or what is a common factor both of those terms that I can divide out and you can see they both share an X so I can take out an X and I can also divide out a two so by dividing out a 2X from both of these terms terms over here I'm going to be left with a 3X and over here I'm going to be left with a pos2 now the goal of factoring by grouping is to get whatever's inside the parentheses to be exactly the same do you see the numbers are exactly the same however these are both positive these are both negative so I need to think of a GCF that I can factor out to take these both negative to be positive and hopefully if you're thinking well why don't you just factor out a negative one well then you would be correct and it'd be like plus technically like let's do this let's rearrange this so technically we have that plus your negative 1 and left over here is now a positive 3x plus 2 well ladies and gentlemen what is plus a negative that's still just going to be a subtraction right so let's rewrite this so I have a 2X 3x + 2 minus1 3x + 2 okay notice that these are both exactly the same just like how we have both had an X here I factored out the X right since these both of these terms which are separated by the subtraction sign I now can factor out the common term that I have here which is going to be a 3x + 2 what is going to be left over a 2X and a Nega 1 so I have a 3x + 2 can be multiplied by a 2X minus one so again whenever you see a polinomial with four terms think Factor by grouping all right the last one is going to be one that students always seem to forget and whenever we see two terms allows to again students think difference of two squares difference of two squares that's what I drill them into their brain however when we're dealing with a cube we actually have the sum or the difference of two cubes so you just have to remember a lot of times students will get confused because they just remember difference of two squares and they always they sometimes only think that the difference of only two cubes is going to be the result but that's not true so the difference of two cubes has a formula that goes like this all we simply need to remember is if we have a cub plus b Cub the first binomial is always going to be symmetric of that just as a plus b if it was a cub minus B Cub then it'd be an A minus B and then over here you're just going to have this change sign so it's a^ s minus a B plus a b squ if it was a 2 or a cub minus B Cub it would be a minus b a 2 + a plus b s okay so what we need to be able to do is identify is this in that format can I rewrite this first first term as something cubed and the last term as something cubed well obviously you see the X Cub so like that's good but we have to make sure we have the number as well so what I want see is can I rewrite eight as a number cubed and hopefully you recognize yeah I can rewrite that as a two right because 2 cubed 2 * 2 * 2 = 8 and then for 27 can I rewrite 27 as a cubed number and the answer is yes of course I can um that's going to be a 3 Cub so if a cubed is equal to a 2X cub and if I want to be able to solve for a we'll take the cube root of both sides a is going to equal a2x and if B Cubed is equal to a 3 cubed then hopefully you recognize B is equal to 3 so now if I know my value of a and I know my value of B then all we're simply going to do guys is plug them into this formula so 2 a + 3 and then we have a 2x^ 2 - x * 3 + A3 2 okay now let's go ahead and simplify and there he goes ladies and gentlemen those are the three most popular pols that I believe you need to know hope this video helps cheers