This is a section from Gilbert Strang's Linear Algebra book:
Although I understand the uniqueness and existence of that matrix, I am having trouble understanding what he means by, "The only such polynomial that vanishes at... No other polynomial of degree $n-1$ can have $n$ roots." What does he mean by vanishing and why does no other polynomial have $n$ roots?
We say that the polynomial $P(t)=x_1+x_2t+\ldots+x_nt^{n-1}$ vanishes at some point $t=t_0$ if $t_0$ is a "root" or a "zero" of that polynomial, i.e. if $P(t_0)=0$.
What the author is trying to say is that a polynomial of degree at most $n-1$ can have at most $n-1$ different zeros (barring the case where the polynomial itself is a zero polynomial - i.e. all its coefficients are zero and therefore it "vanishes" everywhere). This result is a well-known result from algebra, and is a consequence of unique factorization for polynomials.