Let F be a field and let fx gx elementof Fx have degree less

Let F be a field and let f(x), g(x) elementof F[x] have degree less than or equal to n. Suppose that c_0, c_1, c_2, ..., c_n, are distinct elements of F such that f(c_k) = g(c_k) for k = 0, 1, 2, ..., n. Prove that f (x) = g(x) in F[x].

Solution

given that F be a field and f(x),g(x) belong to F[x] hae degree less than or equal to n

now C₀, C₁, C₂, .... are distinct elements of F such that f(C_K) = G(C_K)

for all k = 0,1,2,3,4....

hence,

within given field F

f(x) = g(x) for entire domain

and hence they coincide

hence f(x) = g(x) in F[X]