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Finding Polynomials from their Zeros

  1. Jun 11, 2007 #1
    1. The problem statement, all variables and given/known data
    Prove that x = 2^(1/2)+2^(1/3) is irrational.


    2. Relevant equations



    3. The attempt at a solution

    The hint to the exercise says that I should first show that x satisfies an equation of the form x^6+a_1*X^5+...+a_0 = 0, where the coefficients are integers.

    I am having trouble showing that x satisfies such an equation.

    I tried multiplying (x-2^(1/2)-2^(1/3))(x+2^(1/2)-2^(1/3)) but was not successful in getting integer coefficients.
     
  2. jcsd
  3. Jun 11, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Try putting some more factors in.


    Or just start from

    x=sqrt(2)+cbrt(2)

    and manipulate it to find an expression x satsifies. (eg, if x=sqrt(2), then squaring both sides yields x^2=2, so x satisfies x^2-2=0. n.b. don't square both sides of what you started wit -try something more sophisticated than that).
     
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