If `f(x)=x^3+b x^2+c x+da n df(0),f(-1)` are odd integers, prove that `f(x)=0` cannot have all integral roots.

Explanation

`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.