Show that f(t) has degree n. Find the coefficient of k of $t^n$ using the formula from n-1 case. $$f(t)= det \begin{pmatrix} 1 & 1 & \cdots & 1 \\\\ a_0 & a_1 & \cdots & t \\\\ a_0^2 & a_1^2 & \cdots & t^2 \\\\ \vdots & \vdots & \ddots & \vdots\\\\ a_0^n & a_1^n & \cdots & t^n \\\\ \end{pmatrix} $$ How do I show the degree being equal to n? It seems like it just should be; however, I have no idea how to show this. The formula referred to in the second question is $$det(a)=\Pi_{i>j}(a_i-a_j)$$ I do not know how to apply this to the n-1 case.
2026-04-05 12:08:23.1775390903
Degree of a Vandermonde Matrix
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To see that the degree if $n$, expand along the last column using Laplace expansion. Note that the other columns are independent of $t$.
The coefficient is $(-1)^{n+n}\det(a)$