Abstract Algebra[zero polynomial of infinite field]

[malcmitch20]

What would be the best way to show that if F is an infinite field and f(x) is a polynomial in F[x] and f(a)=0 for an infinite number of elements a of F, that f(x) must be the zero polynomial? It kind of just makes logical sense to me, so I can't think of a way to actually show this. please help

 

[disregardthat]

Use that if f(a) = 0, then x-a divides f(x) in F[x] and show ultimately that any non-zero polynomial have a finite number of zeroes.

 

[malcmitch20]

thanks