If `k_(1)` and `k_(2)` (`k_(1) gt k_(2)`) are two non-zero integral values of `k` for which the cubic equation `x^(3)+3x^(2)+k=0` has all integer roots, then the value of `k_(1)-k_(2)` is equal to_______

Answer

8

Explanation

`x^(3)+3x^(2)+k=0 to alpha, beta, gamma`
`alpha+beta+gamma=-3, alpha beta +beta gamma+gamma alpha=0, alpha beta gamma=-k`
`alpha^(2)+beta^(2)+gamma^(2)=9`
`because alpha,beta, gamma epsilonIimpliesalpha^(2)=9, beta^(2)=0, gamma^(2)=0`
or `alpha^(2)=4, beta^(2)=4, gamma^(2)=1`
`alpha^(2)=4, beta^(2)=4, gamma^(2)=1`
Possible roots: `+-3, 0, 0, +-2, +-2, +-1`
But `alpha beta+beta gamma +gamma alpha=0`
So, possible roots are `3, 0, 0, -3, 0, 0, 2, 2, -1, -2, -2 , 1`
Possible non-zero valuesof `k` are `-4` and 4