Is a polynomial having $d+1$ coefficients related to how it uniquely satisfies $d+1$ points?

Question

A degree $d$ polynomial $p(x)=a_d x^d+a_{d-1} x^{d-1}+\cdots+a_1 x+a_0$ has $d+1$ coefficients. Is this fact related in any meaningful way to how a polynomial uniquely satisfies $d+1$ points with unique $x$-values? (unique in that it is the only polynomial of degree $d$ or less to satisfy these points).

Answer 1

Yes, every point it goes through gives you an equation of the $a_i$'s. After plugging in $d+1$ points, you get a system of $d+1$ equations with unknowns $a_0\dots a_d$, with a unique solution (if you haven't used the same x-value twice), meaning those points determine your polynomial function completely.