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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_______ |
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Answer» `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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