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Polynomials do or don't have integer roots?

  1. Aug 14, 2007 #1
    1. The problem statement, all variables and given/known data

    Is it there a method to find out if a polynomial has no integer roots?


    3. The attempt at a solution
    I tried the division of polynomials, as well as the Horner's Method, but no luck.
     
  2. jcsd
  3. Aug 14, 2007 #2
    Use "rational root theorem".
     
  4. Aug 14, 2007 #3
    Doesn't the rational root theorem only apply to polynomials over the rationals?
     
  5. Aug 14, 2007 #4

    Dick

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    Yes, basically. Is there a method for general real polynomials?
     
  6. Aug 14, 2007 #5
    Povatix,

    Does the polynomial itself have integer coefficients? If so, you can use Eisenstein's criterion.
     
  7. Aug 14, 2007 #6

    Dick

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    Irreducible is sufficient to show there is no rational root, but it's not necessary. Kummer was right. Just check all the possible rational roots, if none are integers, then there are none. The OP's polynomial almost certainly has rational coefficients. And if it doesn't I wish they had displayed it.
     
  8. Aug 20, 2007 #7
    Here's the equation.

    [tex]\chi^{10} + p\chi^{9} - q\chi^{7} + r\chi^{4} - s = 0[/tex]

    thnx in advance
     
  9. Aug 20, 2007 #8

    HallsofIvy

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    If the coefficients are all integers, then the only possible integer solutions are s or its factors. The only way I know of determining whether any of those ARE solutions is to try them in the equation. Are you doing this for specific values of p, q, r, s? Obviously, the equation you give will have integer values for some values of p, q, r, s and not for others.
     
  10. Aug 20, 2007 #9
    Oh right, sorry, the question also states that p q r and s are all odd integers
     
  11. Aug 20, 2007 #10

    HallsofIvy

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    If x is an odd integer, then so is any power of it and so is any odd integer times any power. What can you say about the sum (or difference) of 5 odd integers?

    If x is an even integer, then so is any power of it and so is any integer times any power. What can you say about the sum (or difference) of 4 even integers. In order that the value of the polynomial be 0, what must s be?

    It would have helped a lot if you had given us all this information to begin with!
     
  12. Aug 21, 2007 #11

    Gib Z

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  13. Aug 21, 2007 #12
    Yes, it's true. This is part of a much larger enrichment program aimed at the top students in Year 10, which includes a textbook that is meant to aid you through the 16 questions.

    P.S. How did you find out?
     
  14. Aug 21, 2007 #13

    Gib Z

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    I'm Psychic =]

    Jokes, I did it this year too. You will like that thread then, it has alot of other questions from the program :)
     
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