Taylor Expansion for Polynomials/Order 1
Let $R$ be a commutative ring with unity.
Let $f(X)\in R[X]$ be a polynomial.
Let $a\in R$.
Then there exists a polynomial $g(X)\in R[X]$ such that:
- $f(X+a) = f(X) + a f'(X) +a^2 g(X)$
where $f'$ denotes the formal derivative of $f$.
By linearity, it suffices to prove this for $f(X)=X^n$.
This is now a direct consequence of the Binomial Theorem.