If $a + b + c = 0$, one root of $\left| \begin{array}{ccc} a - x & c & b \\ c & b - x & a \\ b & a & c - x \end{array} \right| = 0$ is

Let the matrix be $M$. Since each row sum is $a+b+c-x$, and given $a+b+c=0$, each row sum is $-x$. So for the vector $\begin{bmatrix}1\\1\\1\end{bmatrix}$, we have $$M\begin{bmatrix}1\\1\\1\end{bmatrix}=-x\begin{bmatrix}1\\1\\1\end{bmatrix}.$$ Thus $-x$ is an eigenvalue of $M$. For the determinant equation $|M|=0$, one eigenvalue must be $0$. That happens when $$-x=0.$$ Hence $$x=0.$$ Now checking the options, this matches option $\text{D}$.