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Simplifying Equations with Polynomial denominators/numerators

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data

    I uploaded a picture of the question because I didn't want to confuse people from the way it looks because it is pretty long.

    http://tinypic.com/r/2itpm39/5

    2. Relevant equations

    -None-

    3. The attempt at a solution

    I went from the original equation and took an 'x' out of the first numerator and then got the reciprocal of the x^n and multiplied that through

    1. Original Equation
    2. (((x^n)-8)/((x^2n)+(2x^n)+1)) x ((x^2n)-(4x^n)-5)/(x^n)

    I don't know what I can factor out from this point..
    Please help and thanks
     
    Last edited: Feb 5, 2012
  2. jcsd
  3. Feb 5, 2012 #2

    SammyS

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    x2n = (xn)2

    That in itself may help you. If not, then let u = xn and substitute that.
     

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  4. Feb 5, 2012 #3
    Do you mean x^2n= (x^2)^n

    Also as an example, if I took x^n-8 and did that, would it be:

    ((x)^3n)-(2)^3
    and i dont know where to go from there

    and for this one:

    (x^2n)-(4x^n)-5
    ((x^2)^n)-((4x)^n)-5
    what would i do next?
     
  5. Feb 5, 2012 #4

    eumyang

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    While that's true, SammyS was right to say
    x2n = (xn)2

    So if you were to make the substitution u = xn,
    xn - 8 would become u - 8.
    With the substitution, it would be
    x2n - 4xn - 5
    = (xn)2 - 4xn - 5
    = u2 - 4u - 5

    Now do the same kind of substitution in the first denominator. Doesn't that expression look familiar now? And can't you do something with u2 - 4u - 5?
     
  6. Feb 5, 2012 #5
    I substituted all the x^n with y's and ended up with:

    (((x^n)-5)((x^n)-2)((x^2n)+(2x^n)+4))/(x^n)((x^n)+1)

    Is that all it can get factored and simplified?
     
  7. Feb 5, 2012 #6
    Also, I think i will lose points by substituting, what is the way without it?
     
  8. Feb 5, 2012 #7

    SammyS

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    ((xn ) - 2) ((x2n ) + (2xn ) + 4) = x3n - 8 ≠ xn - 8
     
  9. Feb 7, 2012 #8
    I redid it and got: (((x^n)-8)((x^n)-5))/(x^n)((x^n)+1)

    Is that correct?

    Sorry, I didn't get what you are trying to say
     
  10. Feb 7, 2012 #9

    eumyang

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    If you mean
    [tex]\frac{(x^n-8)(x^n-5)}{x^n(x^n+1)}[/tex]
    ... then yes, looks right.
     
  11. Feb 7, 2012 #10

    Mentallic

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    Substituting isn't necessary, but makes things look simpler.

    For example, if you were asked to factorize

    [tex]x^2+4ax+4a^2[/tex]

    then it isn't very clear what needs to be done, but if you let 2a=b then you have

    [tex]x^2+2bx+b^2[/tex] and this is clearly a perfect square, so you can factorize it as so:

    [tex](x+b)^2[/tex] and then you can substitute back in at the end to obtain

    [tex](x+2a)^2[/tex]
     
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