Check at $x = 0,$ we get $\left| {\,\begin{array}{*{20}{c}}{{\omega ^2}}&\omega &1\\\omega &{{\omega ^2}}&1\\1&\omega &{{\omega ^2}}\end{array}\,} \right|$

= ${\omega ^2}({\omega ^4} - \omega ) - \omega ({\omega ^3} - 1) + 1({\omega ^2} - {\omega ^2})$

= ${\omega ^2}(\omega - \omega ) - \omega \,(1 - 1) + 0 = 0$ Or

$\Delta = \left| {\,\begin{array}{*{20}{c}}{1 + \omega + {\omega ^2} + x}&\omega &1\\{1 + \omega + {\omega ^2} + x}&{{\omega ^2}}&{1 + x}\\{1 + \omega + {\omega ^2} + x}&{x + \omega }&{{\omega ^2}}\end{array}\,} \right|$
by ${C_1} \to {C_1} + {C_2} + {C_3}$

$ = \,\left| {\,\begin{array}{*{20}{c}}x&\omega &1\\x&{{\omega ^2}}&{1 + x}\\x&{x + \omega }&{{\omega ^2}}\end{array}\,} \right|$, ($\because \,1 + \omega  + {\omega ^2} = 0$)

$= 0$, if $x = 0$.