If $\text{'}\omega \text{'}$ is a cube root of unity, then $\Delta =\left[\begin{array}{ccc}x+1& \omega & {\omega }^2\\ \omega & x+{\omega }^2& 1\\ {\omega }^2& 1& x+\omega \end{array}\right]=$

$C_1\to C_1+C_2+C_3$

$\Delta =\left|\begin{array}{ccc}x& \omega & {\omega }^2\\ x& x+{\omega }^2& 1\\ x& 1& x+\omega \end{array}\right|=x\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ 1& x+{\omega }^2& 1\\ 1& 1& x+\omega \end{array}\right|$

$R_2\to R_2-R_1,R_3\to R_3-R_1$

$\Delta =x\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ 0& x+{\omega }^2-\omega & 1-{\omega }^2\\ 0& 1-\omega & x+\omega -{\omega }^2\end{array}\right|$

$=x[(x+{\omega }^2-\omega )(x+\omega -{\omega }^2)-(1-{\omega }^2)(1-\omega )]=x^3$