# Roots of Polynomial for a Derivative

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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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.