InterviewSolution
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InterviewSolution
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If `f(x)=x^2+b x^2+c x+da n df(0),f(-1)`are odd integers, prove that `f(x)=0`cannot have all integral roots. |
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Answer» `f(0) = d, f(-1) = -1 + b - c + d` `rArr d = odd and - 1 + b - c + d = odd` `rArr b - c = 1 + odd -d` `= (1+ odd) -(odd) = even - odd = odd` (1) Thus, both d and b - c are odd If possible let the three roots `alpha, beta, gamma` be all integers. Now, `alpha beta gamma = - (d)/(1) = -d =` negative odd integers (2) `rArr alpha, beta, gamma` are three integers whose product is odd `rArr alpha, beta, gamma `all are odd Again `alpha+ beta + gamma = - b and alpha beta + beta gamma + alpha gamma = c` (3) `rArr ` b and c both will be odd `rArr (b - c)` will be even which contradicts with (1) Hence, the three roots connot be all integers. |
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