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If `k_(1)` and `(k_(2)gtK+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» Correct Answer - 8 `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 |
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