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Shazailah A.

Precalculus

5 days, 10 hours ago

What is the partial fraction decomposition of x^3+19/(x-1)^2(x^2+4)

University of North Carolina at Chapel Hill

What is the partial fraction decomposition of x^3+19/(x-1)^2(x^2+4)

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Hello. So here we have to find out the partial fraction decomposition off. Two X cube plus two X square plus four X minus three divided by tax square plus X rays to part four You can do Is it gonna be right? This, uh, two X cube plus two X square plus four x My sweet blood by X square Expert Last month on, we can write this impartial function farm like this A by ex Let's be by x square. Yeah, plus C x plus D by expert plus one. Now, if you need to dio to find the partial fraction the competition, you just moved it normally to the right side. Uh, you canceled nominators out and you get something like a x x squared plus one plus B expert plus one plus c x cube class D X square. Now find the and valleys off A B and C. Okay, First, all we gotta equate into this to find the values of A, B, C and D. What I can do is we can put the values affects his boots. So this other only is fun Bill route. And in these equations, that is zero. Because Exco plus one cannot be Zero was real numbers So but actually to the zero. And you get love Aaliyah B, which is gonna be This will become zero this welcome Zero this welcome. Zero So then you get B, which is right my history. Now you have the value of B knock. You need to find the value of a C and D is you can just, uh, combine the powers or approved the power together. So let's just say you're gonna find the valley off right now. Wait. Essay. You're gonna find the value of D so d X squared, plus B. Okay, since there's only two X to the caution of expert here, that means that be experts here, that means that be X square plus D. X square is equal to two X square, right? You cancel the export. That means People's day is to That means B is equal to just I mean, that means he is equal to five. You the same thing for C. So there's X Q here. We actually don't know the value of that. So we do is first of all, we found the value off. We find the value off. Oh yeah. You find the body of a so we can see that there's only one number A x with the culture of X Here, Right on. Northern number on left side. No, the verbal left side has a question has its variable s x So you get a X is equal Thio forex canceled off. Yeah, exile we get A's is equal to four Now you can in front of you See so you can see that a x cube plus c x cube is equal to Thio Excuse you constantly excuse about s four So she has to be modesty. So you get the value of a C. D and B on all you need to do for the partial fraction is just plug these values in and the sun right here. Let me get the answer

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