Let $f\left(x\right)=x^3+a{x}^2+bx+c$ be a polynomial with integer coefficients .If the roots of $f\left(x\right)$ are integer and are in A.P then ‘a’ cannot take the value

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

Roots of $f\left(x\right)$ are $\alpha ,\beta ,\gamma$

$\therefore \alpha ,\beta ,\gamma areinA.P\Rightarrow 2\beta =\alpha +\gamma$

Sum of the roots = -a

$\Rightarrow \alpha +\beta +\gamma =-a$

$\Rightarrow 2\beta +\beta =-a\Rightarrow \beta =\frac{-a}{3}$

It is given that roots are integers

$\therefore$ $\beta$ is an integer when a is  a multiple of 3 option $\left(a\right)\left(c\right)\left(d\right)$ are multiple of 3

$\therefore a\ne 1214$