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Solving a certain polynomial

  1. Oct 28, 2009 #1
    So I have an equation:
    h(x) = [tex]\sum a_{i}x^{i}[/tex] from i=0 to d.

    I know [tex]a_{i}[/tex] and x.
    I am trying to prove that there is a y where g(x) = [tex]\sum a_{i}y^{i}[/tex] from i=0 to d, g(x) = h(x), and y does not = x.

    How do I do this? Sorry for the bad use of Latex.
     
  2. jcsd
  3. Oct 28, 2009 #2

    Office_Shredder

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    In general? You can't, for example, the polynomial x3 is 1-1. It's unclear what you mean here, g(x) is not actually a function of x, and in fact seems to be h(y). So what you really want is to find x and y so that h(x)=h(y) right?
     
  4. Oct 28, 2009 #3
    Yes, sorry. That was a typo. That is g(y), not g(x).
     
  5. Oct 28, 2009 #4

    HallsofIvy

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    It's still not clear what you want to prove. If you have [itex]h(x)= \sum_{i=0}^d a_i x^i[/itex] and you replace the variable x by any y, you get [itex]h(y)= \sum_{i=0}^d a_iy^i[/itex]. It is the same function, just written differently.

    If you mean x and y to be specific numbers and want to prove that there exist [itex]y\ne x[/itex] such that h(y)= h(x), you can't- it is not, in general, true. As Office Shredder says, polynomials can be one-to-one. His example of h(x)= x3/sup] shows that.
     
  6. Oct 28, 2009 #5
    I am looking for a y in terms of [tex]a_{i}[/tex] and [tex]x^{i}[/tex] in a solution that contains no polynomial equation. The most important thing I think is to figure out the properties of the summation of [tex]a_{i}[/tex] (which I don't know at all). I know I can't just divide that out, but there has to be a way to separate it from the summation of y.
     
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