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    Roots of Polynomial for a Derivative

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    An elliptic curve can be written as y^2=x^3+ax+b. I need a proof for why x^3+ax+b either have 3 real roots or 1 real root and 2 complex roots. I don't have anything that I know about it prior to asking for help here at Brainmass.

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    https://brainmass.com/math/calculus-and-analysis/roots-polynomial-derivative-630061

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    Problem:

    An elliptic curve can be written as y^2=x^3+ax+b. I need a proof for why x^3+ax+b either have 3 real roots or 1 real root and 2 complex roots.

    Solution:

    It is important first to point out that the coefficients a and b are real, otherwise the given polynomial will have only complex roots. In this case, one needs to analyze the polynomial function
    where ( 1)
    One first remark: ( 2)
    Since f(x) is a continuous function, it means that there exists at least one point for which . ...

    Solution Summary

    An analysis of the nature of the roots of a given polynomial is presented using the properties of the derivative. An elliptic curve proof and real roots are examined.

    $2.19