Determinant of a matrix depending of a parameter

Question

Let's consider a $\left( n+1\right) \times \left( n+1\right)$ matrix: $$\begin{bmatrix} 0^{0}, & 0^{1}, & 0^{2},\ldots & & ,0^{n-1}, & x^{0} \\ 1^{0}, & 1^{1}, & 1^{2},\ldots & & ,1^{n-1} & x^{1} \\ 2^{0}, & 2^{1}, & 2^{2},\ldots & & ,2^{n-1} & x^{2} \\ \ldots \ldots\ \\ n^{0}, & n^{1}, & n^{2},\ldots & & ,n^{n-1}, & x^{n} \end{bmatrix} $$ The question is when its determinant is equal to zero. The trivial answer is $x=1$, but what about other cases? I am looking for a trick to show that it never vanishes except for $x=1$. Does anyone have an idea?