Taylor Expansion for Polynomials/Order 1

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Theorem

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 $f$.


Proof

By linearity, it suffices to prove this for $f(X)=X^n$.

This is now a direct consequence of the Binomial Theorem.

$\blacksquare$