Definition:Polynomial Addition/Polynomial Forms

Definition
Let:
 * $\displaystyle f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
 * $\displaystyle g = \sum_{k \mathop \in Z} b_k \mathbf X^k$

be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

The operation polynomial addition is defined as:
 * $\displaystyle f + g := \sum_{k \mathop \in Z} \left({a_k + b_k}\right) \mathbf X^k$

The expression $f + g$ is known as the sum of $f$ and $g$.

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