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# Intro to partial fraction expansion

## Video transcript

let's see if we can learn a thing or two about partial fraction expansion or sometimes it's called partial fraction decomposition partial fraction expansion expansion the whole idea is to take rational functions in a rational function is just a function or expression where it's one expression divided by another and to essentially expand them or decompose them into simpler parts and the first thing you got to do before you can even start the actual partial fraction expansion process is to make sure that the numerator has a lower degree than the denominator and the the situation the the problem that I've drawn right here I've written right here that's not the case the numerator has the same degree as the denominator so the first step we want to do to simplify this and to get it to the point where the numerator has a lower degree than the denominator is to do a little bit of algebraic division and I've done a video on this but it never hurts to get a review here so to do that we divide the denominator into the numerator to figure out the remainder so we divide x squared minus 3x minus 40 into x squared minus 2x minus 37 so how many times you look at the highest degree term so x squared goes into x squared one time one times this whole thing is x squared minus 3x minus 40 and now you want to subtract this from that to get the remainder and see if I'm subtracting so I'm going to subtract and then minus minus that's a plus plus and then you can add them these cancel out minus 2x plus 3x that's X minus 37 plus 40 that's plus 3 so this expression up here can be re-written as let me scroll down a little bit as 1 plus X plus 3 over x squared minus 3x minus 40 it's important key this might seem like some type of magic thing I just did but this is no different than what you did in the fourth or fifth grade where you learned how to how to convert improper regular fractions into mixed numbers if I had let me just do a little side example here if I had 13 over 2 and I want to turn it into a mixed number what you do you probably do this in your head now but what you did is you divide the denominator into the numerator just like we did over here 2 goes into 13 you say 2 goes into 13 6 times 6 times 2 is 12 you subtract that from that you get a remainder of 1 2 doesn't go into 1 so that's just the remainder so if you wanted to rewrite this it would be the number of times the denominator goes into the numerator that's 6 plus the remainder over the denominator plus 6 plus 1 over 2 and when you did it in elementary school you just write 6 and 1/2 but 6 and 1/2 is the same thing as 6 plus 1/2 and that's exactly the same thing we did here the denominator went into the numerator one time and then there was a remainder of X plus 3 left over so it's 1 plus X plus 3 over this expression now we see that the numerator in this rational expression the numerator does have a lower degree than the denominator right the highest degree here is 1 the highest degree here is 2 so we're ready to commence our partial fraction decomposition and all that is is taking this expression up here and turning it into two simpler expressions where the denominators are the factors of this lower term so given that let's factor this lower term so see what two numbers add up to minus 3 and when you multiply them you get minus 40 it's a C they have to be different signs because when you multiply them you get a negative so it has to be minus 8 and plus 5 so we can rewrite this up here as I'll switch colors 1 plus X plus 3 over X plus 5 times X minus 8 right 5 times 8 is minus 45 times negative 8 is minus 40 plus 5 minus 8 is minus 3 so we're all set now I'll just focus on this part right now we can just have we can just remember that that one is sitting out there out front this is the expression we want to decompose or expand we're going to expand it just to simpler expressions where each of these are the denominator and I will make the claim and if if though if the numbers work out then the claim is true I'll make the claim that I can expand this or decompose this into two fractions where the first fraction is just some number a over the first factor over X plus five plus some number B over the second factor over X minus eight that's my claim and if I can solve for a and B in a way that it actually does add up to this and I'm done and I will have hat fully decomposed this fraction I guess is the way I don't know if even that's the correct terminology so let's try to do that so if I were to add these two terms what do I get when you add anything you find the common denominator and the common denominator the easiest common denominator is to multiply the two denominators so let me write this here so a over X plus five plus B over X minus eight is equal to well let's get the common denominator it's equal to X plus five times X minus 8 and then the a term we would a over X plus five is the same thing as a times X minus eight over this whole thing right I mean if I just wrote this right here and you just cancel these two terms out and you would get a over X plus five and then you could add that to get the common denominator X plus 5 times X minus eight and it would be B times X plus five right important to realize that look the this term is the exact same thing as this term if you just cancel the X minus eight out and this term is the exact same thing as this term if you just cancel the X plus five out but now that we have an actual common denominator we can add them together so we get let me just write the left side here over a over X plus five oh sorry I know I want to write this over here I want to write X plus 3 over X plus 5 times X minus 8 is equal to is equal to the sum of these two things on top a times X minus 8 plus B times X plus 5 all of that over their common denominator X plus 5 times X minus 8 so the denominators are the same so we know that this when you add this together you have to get this so if we want to solve for a and B let's just set that equality we can ignore the denominators so we can say that X plus 3 is equal to a times X minus 8 plus B times X plus 5 now there's two ways to solve for a and B from this point going forward one is the way that I was actually taught when you know in the seventh or eighth grade which tends to be take a little longer then there's a fast way to do it and it never hurts to do the fast way first if you want to solve for a let's pick an X that'll make this term disappear so what X would make this term disappear well if I say X is minus five then this becomes zero and then the B disappears so if X is if we say X is minus five I'm just picking an arbitrary X to be able to solve for this then this would become minus five plus three let me just write it out minus five plus three is equal to a times minus five minus eight let me just write it out minus five minus eight plus B times minus five plus five and I pick the minus five to make this expression zero so then you get pick a brighter color minus five plus three is minus two is equal to what is this minus thirteen a plus this is zero right that's zero minus five plus five zero zero times B is zero and then you divide both sides by minus 13 you get well negatives cancel out you get two over is equal to a and now we can do the same thing up here and get rid of the a terms by making X is equal to 8 if X is equal to 8 you get X plus 3 is equal to 11 is equal to a times 0 a times 0 plus B times what's 5 8 plus 5 is plus B times 13 if there B looks a bit like a 13 and then you get 11 is equal to 13 B divide both sides by 13 you get B is equal to 11 over 13 so we were able to solve for our A's and our B's and so we can go back to our original equation and we could say wow this just has to be equal to 2 over 13 and this just has to be equal to 11 over 13 so if I have if I so this our original our very original thing we wrote up here can be decomposed into one that's this one over here Plus this which is 2 over 13 I'll just write it like this for now 2 over 13 over X plus 5 you could bring the 13 down here if you want to write it so you don't have a fraction over a fraction plus plus 11 over 13 plus 11 over 13 times over X minus 8 and once again you could bring the 13 down so you don't have a fraction over a fraction but we have just successfully decomposed this pretty I don't you know I don't want to say that we necessarily simplify it because you could say oh we only have one expression here now I have three but I've reduced the degree of both the numerators and the denominators and you might say well Sal why would I ever have to do this and you're right in algebra you probably won't but this is actually a really useful technique later on when you get to calculus and actually differential equations because a lot of times it's much easier and I'll throw out a word here that you don't understand to take the integral or the antiderivative of something like this then something like this and later when you do inverse Laplace transforms and differential equations it's much easier to take an inverse applause transform of something like this then something like that so anyway hopefully I've given you another toolkit in your or another tool in your tool kit and I'll probably do a couple more videos because we haven't exhausted all of the examples that we could we could show for partial fraction decomposition