💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

JH
Numerade Educator

Like

Report

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

$$-\frac{1}{a^{n} x}-\frac{1}{a^{n-1} x^{2}}-\dots-\frac{1}{a x^{n}}+\frac{1}{a^{n}(x-a)}$$

Discussion

You must be signed in to discuss.

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.