Let f ( x ) = x 3 + a x 2 + b x + c be polynomial with integer coefficients. If the roots of f ( x ) are…

Given that, $f\left(x\right)=x^3+a{x}^2+bx+c$

Now, since the roots of the polynomial are in Arithmetic Progression,

Let the roots are $s-t, s, s+t$

$\Rightarrow s-t+s+s+t=\frac{-a}{1}$

$\Rightarrow -3s=a$

Hence, $a$ must be multiple of $3$.

From given options we see that only $1214$ is not multiple of $3$.