Let $L \subset \mathbb{P}^n$ be a line.
This is from Joe Harris's Algebraic Geometry, the paragraph right before exercise 1.3.
...it's not hard to see that a polynomial $F(Z)$ of degree $d - 1$ or less that vanishes on $d$ points $p_i \in L$ will vanish identically on L.
Why is this true? I am trying to think of $L$ as a plane in the affine space $\mathbb{A}^{n+1}$. This is pretty obvious to me for $d=2$, but I'm not sure how to show this for higher degrees. I don't think induction would work.
If you plug a standard parametric equation for the line into $F$, you get a polynomial in one variable of degree $d-1$ or less. It is well known that such a polynomial can have at most $d-1$ roots, unless it is the zero polynomial.