Thread: polynomials

Please help. Firstly the wording of n+1 distinct values of x confuses me. Anyway

Prove that if 2 polynomials of degree n, say f(x) and g(x) are equal for n+1 distinct values of x, then the 2 polynomials are equal.

I have:
f(x)=anx^n+ an-1x^n-1 +...+ax+a0

g(x)= bnx^n+bnx^n-1 + ... + bx+b0


then i subtracted g(x) from f(x)

For example if n= 1, saying that f and g are equal for n+1= 2 distinct values means that there are two distinct numbers, a and b, such that f(a)= g(a) and f(b)= g(b). It does NOT imply that there might not be more values of x for which f and g are equal.

Similarly, if n= 2, saying that f and g are equal for n+1= 3 distinct values means that there are thre distinct numbers, a, b and c, such that f(a)= g(a), f(b)= g(b), and f(c)= g(c). Again, it does NOT imply theat there might not be more values of x for which f and g are equal.

You subtracted g(x) from f(x)? Great. That will be a polynomial of degree at most n which is equal to 0 at n+ 1 distinct points. What is the maximum number of distinct zeros a polynomial of degree n can have?