Find all polynomial $f$ such that $f(x)+f(1-x)=1$

Question

While working on a functional equation, I came across this question,

Find all polynomial $f$ such that $$f(x)+f(1-x)=1$$

Clearly there is an infinite number of solution. If you set $$f(x)=\sum_{i=0}^na_i x^i$$ Then $$2a_0+\sum_{i=1}^na_i=1$$ I think this condition is enough for $f$ to satisfy the above functional equation, but can we do better? Can we find $f$ explicitly.

Answer 1

Hint: $f(x + 1/2) - 1/2$ is odd.