Find the partial fraction decomposition of the rational function. 5x^2 + 8/x3 + x2

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Find the partial fraction decomposition of the rational function. 5x^2 + 8/x3 + x2

Mathematics
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Is this \(\frac{5x^2+8}{x^3+x^2}\)?
yes
got a 5x^2 + 8 = A/x^2 + B/x + 1 A(x+1) + B(x^2) (Ax+A) + (Bx^2) (Ax + Bx^2) + A 5x^2 + 8 = (A + B) x + (A)x^2 5x^2+x+8=(A+B) + (A)

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but i think im wrong
the x^2 is throwing me off
So the problem is in how you split it up in the first step. First you've got to realize that the denominator splits into a factor that gets taken into a power. Namely, \(x^3+x^2\) becomes \(x^2(x+1)\), where \(x\) is a factor taken to the second power. The split up thus goes as follows: \[ \begin{align*} \frac{5x^2+8}{x^3+x^2} &= \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}\\ 5x^2+8 &= Ax(x+1) + B(x+1) + Cx^2 \end{align*} \]
where did the A/x come from?
So the \(\frac{A}{x}\) comes from the fact that the \(x\) factor is taken to a power. As another example, if we had taken \(\frac{1}{(x+1)^2}\), the decomposition would split into \(\frac{A}{x+1} + \frac{B}{(x+1)^2}\).
oh ok
Can you solve the problem from there, then?
im trying it
shouldnt the next step be 5x^2 + 8= (Ax^2+Cx^2) + (Ax+Bx) +B? After combining terms?
Exactly.
Ok thanks for your help!
Is -7/x + 8/x^2 + 12/x+1 the right answer?
I got a different answer from that. You can see that something is off because \(A+B\) should equal \(0\), as there is no \(x\) term on the left hand side.
im lost.
So how did you go about solving once you combined terms?
I'll show you
5x^2+8=(A+C)x^2 + (A+B)x + B 5x^2 +0x+8=(A+C) + (A+B) + B then i put it on the matrix function on my calculator and it gave me -7, 8 and 12.
thats how i got Is -7/x + 8/x^2 + 12/x+1.
You probable plugged some numbers in wrong. I get -8, 8, and 13.
probably*
let me see if that works.
Thats it. I dont understand what i did arong though?
I had 1 0 1 5 1 1 0 1 0 1 0 8 as my matrix.
Row 2, column 4 should be a 0, not a one, as there's no \(x\) term.
Ah i see the 0x term stays zero and not 1.
Exactly.
Alright I have now. Thank you so much for helping me. I really appreciate it!
No problem, glad you understood it!

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