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Rationally Integers II (Posted on 2013-12-21)

Determine all possible values of an integer N such that each of the roots of the equation 3X³ - 3X² + N = 0 is a rational number.

No Solution Yet Submitted by K Sengupta

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Solution

Comment 3 of 3 |

Let F(x) = 3x^3 - 3x^2 + N

The first derivative of the polynomial is F'(x) = 9x^2 - 6x. Then F(x) has relative maximum at x=0 and relative minimum at x=2/3. Then 0<=N<=2/3 is necessary for F(x) to have three real roots.

The only integer in the range is N=0, which yields F(x) = 3x^3 - 3x^2 with rational roots {1,0,0}.

Posted by Brian Smith on 2017-06-30 21:47:57

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