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How to compute the following determinant?

$$\left| \begin{matrix} 1 & x_0 & x_0^2 & \ldots & x_0^n \\ 1 & x_1 & x_1^2 & \ldots & x_1^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n ^2 & \ldots & x_n^n \end{matrix} \right|$$

My guess is that it would be: $\prod_{i > j} (x_i - x_j)$ (after carrying it out for $n=2,3$).

I need this to prove that $\Bbb P_n \ni P \mapsto (P(x_0),...,P(x_n)) \in \Bbb R^{n+1}$ is a bijection for distinct $x_i$'s.

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Hint see vandermonde determinant.

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