Definition:Polynomial Addition/Polynomial Forms

Definition
Let $\displaystyle f = \sum_{k \in Z} a_k \mathbf X^k$, $\displaystyle g = \sum_{k \in Z} b_k \mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

Their sum is defined to be:


 * $\displaystyle f + g := \sum_{k \in Z} \left({a_k + b_k}\right) \mathbf X^k$.

It follows from Polynomials Closed under Addition that $f + g$ is a polynomial.