anonymous
  • anonymous
Find the partial fraction decomposition of the rational function. 5x^2 + 8/x3 + x2
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
Is this \(\frac{5x^2+8}{x^3+x^2}\)?
anonymous
  • anonymous
yes
anonymous
  • anonymous
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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anonymous
  • anonymous
but i think im wrong
anonymous
  • anonymous
the x^2 is throwing me off
anonymous
  • anonymous
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: \[ \begin{align*} \frac{5x^2+8}{x^3+x^2} &= \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}\\ 5x^2+8 &= Ax(x+1) + B(x+1) + Cx^2 \end{align*} \]
anonymous
  • anonymous
where did the A/x come from?
anonymous
  • anonymous
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 \(\frac{A}{x+1} + \frac{B}{(x+1)^2}\).
anonymous
  • anonymous
oh ok
anonymous
  • anonymous
Can you solve the problem from there, then?
anonymous
  • anonymous
im trying it
anonymous
  • anonymous
shouldnt the next step be 5x^2 + 8= (Ax^2+Cx^2) + (Ax+Bx) +B? After combining terms?
anonymous
  • anonymous
Exactly.
anonymous
  • anonymous
Ok thanks for your help!
anonymous
  • anonymous
Is -7/x + 8/x^2 + 12/x+1 the right answer?
anonymous
  • anonymous
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.
anonymous
  • anonymous
im lost.
anonymous
  • anonymous
So how did you go about solving once you combined terms?
anonymous
  • anonymous
I'll show you
anonymous
  • anonymous
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.
anonymous
  • anonymous
thats how i got Is -7/x + 8/x^2 + 12/x+1.
anonymous
  • anonymous
You probable plugged some numbers in wrong. I get -8, 8, and 13.
anonymous
  • anonymous
probably*
anonymous
  • anonymous
let me see if that works.
anonymous
  • anonymous
Thats it. I dont understand what i did arong though?
anonymous
  • anonymous
I had 1 0 1 5 1 1 0 1 0 1 0 8 as my matrix.
anonymous
  • anonymous
Row 2, column 4 should be a 0, not a one, as there's no \(x\) term.
anonymous
  • anonymous
Ah i see the 0x term stays zero and not 1.
anonymous
  • anonymous
Exactly.
anonymous
  • anonymous
Alright I have now. Thank you so much for helping me. I really appreciate it!
anonymous
  • anonymous
No problem, glad you understood it!

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