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Rationally Integers II (Posted on 2013-12-21) Difficulty: 3 of 5
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 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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