Determine all possible values of an integer N such that each of the roots of the equation 3X^{3} - 3X^{2} + N = 0 is a *rational number*.

(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 **

** non-****integer**

but rational roots fitting

a **non-****integer coefficient N**

*Edited on ***December 22, 2013, 12:44 pm**