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Smallinomial (Posted on 2013-09-25) Difficulty: 2 of 5
The cubic equation ax3+bx2+cx+d=0 has non_zero integer coefficients and distinct integer solutions. Find the smallest possible value of |a|+|b|+|c|+|d|

No Solution Yet Submitted by Danish Ahmed Khan    
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analytical solution Comment 2 of 2 |

the absolute minimum possible value for
|a|+|b|+|C|+|d| is 4, Ady showed an example
for |a|+|b|+|c|+|d|=6, so now all that is
left is to determine if a solution exists
for 4 or 5.

|a|+|b|+|c|+|d|=4
since all are non-zero this requires
a,b,c,d each to be either +-1
thus d can be either 1 or -1
the product of the three integer zeros
is equal to d, however neither 1 or -1
can be factored into 3 distinct integer
factors.  Thus this absolute minimum is
not possible.


|a|+|b|+|c|+|d|=5
since all are non-zero this requires
3 of them to be equal to +-1 and one to be
equal to +-2.  Thus d is either +-1 or +-2
we already ruled out +-1 above, so that leaves.
The only way to factor 2 into 3 integers is (1,1,2)
in order for them to be distinct the two 1's must be
of opposite signs, thus giving us either (-1,1,2)
or (-2,-1,1).  (-1,1,2) was already covered by
Ady thus that just leaves (-2,-1,1) which gives us
(x-2)(x-1)(x+1)=x^3-2x^2+x-2 which does not
give us the desired absolute sum of 5.
Thus it is not possible to achieve an asolute
sum of 5 either.  Thus 6 must be minimum sought.


  Posted by Daniel on 2013-09-25 14:02:20
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