Talk:Polynomials Closed under Addition


 * Since $\displaystyle f = \sum_{k\in Z}a_k\mathbf X^k$ and $\displaystyle g = \sum_{k\in Z}b_k\mathbf X^k$ are polynomials, there are $k_{min}$ and $k_{max}$ such that $k \notin [k_{min}..k_{max}]$ implies $a_k = b_k = 0$.

$\implies$
 * $k \notin [k_{min}..k_{max}]$ implies $a_k + b_k = 0$.

$\implies$
 * $\displaystyle f \oplus g = \sum_{k \in Z}(a_k + b_k) \mathbf X^k = \sum_{k = k_{min}}^{k_{max}}(a_k + b_k) \mathbf X^k$

$\implies$
 * $\displaystyle f \oplus g$ is a polynomial.