Polynomial of $n+1$ Distinct Value and Uniqueness.

Question

Let $F$ be a field and $f, g ∈ F[x]$ two polynomials of degree $n$ over $F$ . Suppose that there exist $n + 1$ distinct values $α_i ∈ F$ , such that $f(α_i) = g(α_i)$ for all $i$.

How can I prove $f = g$?

Can I find the proof in any book?

Answer 1

A polynomial $h$ of degree $n$ over a field $F$ is either the zero polynomial or has at most $n$ roots. In your case, $h=f-g$ has degree $m\le n$ and has more than $n$ roots. Hence $h=0$ and thus $f=g$.

Reference: Polynomial of Degree $n$ over a field has at most $n$ zeros, counting multiplicity

Answer 2

Hint:

$\alpha_i$ is a root of $f-g$ and hence $x-\alpha_i$ divides $f-g$.