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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 re: the trivial ones | Comment 2 of 3 |
(In reply to the trivial ones by Ady TZIDON)

Following a lengthy process of solving by a plug-in method

got  the following condition to get three real roots

q=N*(9N-4)  Should be  a non-positive number

N=0  roots 1,0,0

N=4/9 is not an integer, and so are the numbers  between
0 and 4/9, the only numbers that would make q negative.

Ergo: N=0 is the only integer answer

For n=4/9
the equation may be factored into 

( x+1/3)*(x-2/3)^2=0

and the roots :     -1/3, 2/3, 2/3  
 but rational roots   fitting
non-integer coefficient N

Edited on December 22, 2013, 12:44 pm
  Posted by Ady TZIDON on 2013-12-22 04:46:36

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