Talk:Polynomials Closed under Addition

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

$\leadsto$

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

$\leadsto$

$\ds f \oplus g = \sum_{k \mathop \in Z} \paren {a_k + b_k} \mathbf X^k = \sum_{k \mathop = k_{min} }^{k_{max} } \paren {a_k + b_k} \mathbf X^k$

$\leadsto$

$f \oplus g$ is a polynomial.