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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»


Solution :`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`
`becausealpha,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


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