all right i'm going to answer the biggest why when we are doing partial fractions take a look and that's the case when we are dealing with a big fraction like this when we have a repeating factor on the bottom and let's take a look at x plus 2 squared break this apart okay yes we see the x plus 2 squared right here for this little fraction already but why do we also need to put down this x plus 2 to the first power right here and as i said this right here is perhaps the biggest why when we are doing partial fractions and before i answer this question let me just go over the setup with you guys from where doing partial fractions take a look so here is the first case before anything though i should also mention that in fact the top doesn't matter as long as the degree on the top is less than the degree on the bottom so this could have been 3x minus 11 or 2x minus 217 the setup will still be the same as long as the degree is strictly less than the degree on the bottom the degree here is one the degree here is one plus two which is three so the setup will be like this and let's just use the same numerator 2x plus 1. so if we take a look at 2x plus 1 over if we just have x plus 1 times x plus 2. this is nice this is everybody's favorite because we can just break them apart with x plus one right here and then the second fraction we get x plus two notice that they are just both linear meaning degree one so the top will be just constants because you always want to make sure when you set up the degree on the top is one less than the degree on the bottom right so degree is one that means the top right here is degree zero meaning just constants and then we can talk about how to get the a and b by using the kapha method in another video but this is the setup first when we have two different factors and they are both linear now let's take a look at the second one what if we have a quadratic let's say 2x plus one on the top over x plus one but this is x squared plus two this right here is a quadratic it's kind of similar to this because technically this is also quadratic but the thing is that this right here it's actually a irreducible quadratic meaning that we cannot factor this anymore with real numbers so we just have to deal with this right here but technically this right here is what means x plus 2 times x plus 2. this right here is a repeating factor so i'll just say this is the repeating factor and this right here we can look at it as x plus 2 times x plus 2. so when you have a repeating factor like this you have a power on the outside of the parentheses and then you will have to do things like that but let's talk about them later how do we take care of our irreducible quadratic have a look again the first fraction will have x plus one on the bottom and then for the second fraction we'll just use this that's x squared plus 2. on the top here it will just be a constant but for this right here because it's a quadratic on the bottom and the top has to be one degree less than the bottom so we will have to set up a linear and you have to make sure you put on the general form so we get bx plus c done and of course to solve for abc it's going to take some time but maybe another video all right number three check this out i will still give you a quadratic but this one let's say we have 2x plus 1 over x plus 1 here and here i will just multiply it with x squared this is also quadratic but this quadratic is similar than this because this right here means what x times x all right okay let's use the same strategy and how we set this up earlier i'm going to first write down the x plus 1 here with just a constant on the top but for this one okay i will just keep it as how the is which is x squared but at the top will be a linear meaning bx plus c but we recognize what we can do with this part see the bottom here has just one term so that we can actually just split the fraction and we will get okay still a over x plus 1 but for the second part we can just write this as b x over x squared first and then c over x squared and what can we do right here yes we can cancel cancel so ladies and gentlemen we will end up with a over x plus one and then plus b over x and then plus c over x squared as you can see right here we have x squared and please look at this x squared as x times x so we can put the square on outside it's a repeating factor and when we set this up yes we will have the x squared here but we also need x to the first power here i call this build up the power what it means is that you start with x to the first power and then you build up just keep adding one until you reach this power imagine if you have let's say let's say we have 2x plus 1 over still x plus 1 but let's say we have x to the first power you don't need to do this step every single time just remember build up the power and set it up correctly firstly we will have x plus 1 and then the top will be a constant let's say a okay x to the fourth power so here we go start with x to the first power and then we just keep going x squared and then x to the third power and then x to the fourth power this right here is linear so that means the top will be a constant b and then the top will stay the same kind so another constant another constant another constant all right so that's why we do that but of course you would be thinking that this is x squared this is just x it might be different don't worry i got you this is how you can explain this right here so have a look we still want to answer this expression what we can do is just do some substitution and let's just say that t and i will just say that t equal to this so that we will end up with similar form so i'll just say t equals x plus 2. and then we can do a few things right because for example we can say this is the same thing x is equal to t minus 2 and then we can say x plus 1 just add 1 on both sides so x plus 1 will be t minus one and then we also need the top which is two x plus one okay let's do this in our head so this will be two times t which is two t and then two times minus two is minus four and then plus one is minus three okay so we can look at this expression as what the top is two t minus three so let's write that down and then over x plus 1 is that t minus 1 and then the x plus 2 is t so we have the t squared and now per whole discussion earlier this right here would just be what t minus 1 even though we have the plus one but again it's the form it's not about the number this right here is t minus 1 this linear the top will be a constant and likewise this right here will be okay build up the power t to the first and then t to the second the top will just be a constant likewise the top will be a constant we're all done huh and of course in the end we can set this back just to match it so ladies and gentlemen a over t minus one is x plus one so we can put that down and then b and then t is x plus 2. and of course lastly we have that c over x plus 2 and then squared so ladies and gentlemen this is the reason why that we must have this term right here hopefully this video helps for most tutorials for more tutorials for more tutorials