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anonymous

  • 5 years ago

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

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  1. anonymous
    • 5 years ago
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    Is this \(\frac{5x^2+8}{x^3+x^2}\)?

  2. anonymous
    • 5 years ago
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    yes

  3. anonymous
    • 5 years ago
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    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)

  4. anonymous
    • 5 years ago
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    but i think im wrong

  5. anonymous
    • 5 years ago
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    the x^2 is throwing me off

  6. anonymous
    • 5 years ago
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    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: \[ \begin{align*} \frac{5x^2+8}{x^3+x^2} &= \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}\\ 5x^2+8 &= Ax(x+1) + B(x+1) + Cx^2 \end{align*} \]

  7. anonymous
    • 5 years ago
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    where did the A/x come from?

  8. anonymous
    • 5 years ago
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    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 \(\frac{A}{x+1} + \frac{B}{(x+1)^2}\).

  9. anonymous
    • 5 years ago
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    oh ok

  10. anonymous
    • 5 years ago
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    Can you solve the problem from there, then?

  11. anonymous
    • 5 years ago
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    im trying it

  12. anonymous
    • 5 years ago
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    shouldnt the next step be 5x^2 + 8= (Ax^2+Cx^2) + (Ax+Bx) +B? After combining terms?

  13. anonymous
    • 5 years ago
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    Exactly.

  14. anonymous
    • 5 years ago
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    Ok thanks for your help!

  15. anonymous
    • 5 years ago
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    Is -7/x + 8/x^2 + 12/x+1 the right answer?

  16. anonymous
    • 5 years ago
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    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.

  17. anonymous
    • 5 years ago
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    im lost.

  18. anonymous
    • 5 years ago
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    So how did you go about solving once you combined terms?

  19. anonymous
    • 5 years ago
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    I'll show you

  20. anonymous
    • 5 years ago
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    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.

  21. anonymous
    • 5 years ago
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    thats how i got Is -7/x + 8/x^2 + 12/x+1.

  22. anonymous
    • 5 years ago
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    You probable plugged some numbers in wrong. I get -8, 8, and 13.

  23. anonymous
    • 5 years ago
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    probably*

  24. anonymous
    • 5 years ago
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    let me see if that works.

  25. anonymous
    • 5 years ago
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    Thats it. I dont understand what i did arong though?

  26. anonymous
    • 5 years ago
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    I had 1 0 1 5 1 1 0 1 0 1 0 8 as my matrix.

  27. anonymous
    • 5 years ago
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    Row 2, column 4 should be a 0, not a one, as there's no \(x\) term.

  28. anonymous
    • 5 years ago
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    Ah i see the 0x term stays zero and not 1.

  29. anonymous
    • 5 years ago
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    Exactly.

  30. anonymous
    • 5 years ago
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    Alright I have now. Thank you so much for helping me. I really appreciate it!

  31. anonymous
    • 5 years ago
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    No problem, glad you understood it!

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