Conjecture $1:$ There is exactly one polynomial of degree $n-1$ passing through $n(\in\mathbb{N})$ points in $\mathbb{R}^2.$
Conjecture $2:$ There are infinitely many polynomials of degree $n$ passing through $n(\in\mathbb{N})$ points in $\mathbb{R}^2.$
These two statements are easy to prove for $n=1,$ and $n=2.$ But, for $n≥3,$ the proof method I'm using gets a little tedious. Basically, I formed equations and solved them to get the values of the coefficients. Is there any way to prove this for a general $n\in\mathbb{N}?$
Note: I am only considering polynomials in one variable that have real coefficients, and accept real inputs. Moreover, when I say that $p,$ a polynomial passes through $(x_0,y_0)\in\mathbb{R}^2,$ I mean that $p(x_0)=y_0.$ The points I'm considering have distinct $x$ coordinates.