Problem 75 Hard Difficulty

If $ a \neq 0 $ and $ n $ is a positive integer, find the partial fraction decomposition of
$$ f(x) = \frac{1}{x^n (x - a)} $$
[Hint: First find the coefficient of $ \frac{1}{(x - a)} $. Then subtract the resulting term and simplify what is left.]

Answer

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Video Transcript

Let's find the partial fraction in opposition. And here let's not that we're given is non zero. So here we can rewrite this. So for the first term X event, let's use what the author calls case, too. This is when you have repeated linear factors. So then a two over X squared and we go all the way up xn and then we'LL have a case One. This is a non repeatedly in your factor X minus a with good and multiply And both sides by the denominator on the left So on the right hand side and so on and we'LL have all the way up to a and X minus a and then we'LL have be accident Now the next step would be to go ahead and combine depending on the power of X So flu pullout xn one a one plus me Pull out a xn minus one. You get a two minus a one and so on. And when we plotted X day and minus a a and minus one and then finally the constant term and a So now we have comparing the constant terms We have a one on the left. We have a negative and and the right So that gives us a N equals negative one over, eh? And for instance, we can use that to find this. We know this is equal to zero because there's no X on the left. So we have a end here, Let me write it this way. And minus one equals a A and over, eh equals negative. A one over a square. Similarly, a N minus two equals a and minus one. Over, eh? So here, negative one. Over a cute. That's a negative up there and continuing in this fashion level. Me Go on to the next. Actually, let me take a step back here. That should have been in minus one. So what I had. Okay, so let me go. The next page, a one is a two over, eh? A three over a square. Just using the same formula repeatedly. Negative one over eight, then and then be equals negative. A one, because a one plus zero zero zero and then this is positive. One over. And so now that we've found all of our coefficients for the partial fraction to composition, just plug a moment. Negative one. An ex those are capital that was a one and then a two and minus one X squared and so on. Great. And then finally negative. One of rain and exit and and then R B was won over eight of the end, and that's her room was X minus a. There it is. That's our final answer.