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Math Help - 2 polynomial problems

  1. #1
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    2 polynomial problems

    problem 1
    let f(x) and g(x) be two quadratic polynomials all of whose coefficients are rational numbers.suppose f(x) and g(x) have a common irrational root.show that g(x)=r.f(x),for some rational number r.

    problem 2
    suppose the roots of x^2+px+q = 0 are rational numbers and p and q are integers.
    then show that the roots are integers.

    please help!and please give the problem number to your answers
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by earthboy View Post
    problem 1
    let f(x) and g(x) be two quadratic polynomials all of whose coefficients are rational numbers.suppose f(x) and g(x) have a common irrational root.show that g(x)=r.f(x),for some rational number r.
    For such a polynomial the irrational roots occur in conjugate pairs, if such quadratics share one irrational root they have identical roots.

    CB
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by earthboy;563081[B
    problem 2
    [/B]suppose the roots of x^2+px+q = 0 are rational numbers and p and q are integers.
    then show that the roots are integers.
    Because this is a monic polynomial the rational roots theorem gives the required result immediately

    CB
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