Multiplication of Polynomials Distributes over Addition

Theorem
Multiplication of polynomials is left- and right- distributive over addition.

Proof
Let $(R, +, \circ)$ be a  commutative ring with unity.

Let $\left\{{X_j: j \in J}\right\}$  be a set of indeterminates.

Let $Z$ be the set of all multiindices indexed by $\left\{{X_j: j \in J}\right\}$.

Let


 * $\displaystyle f = \sum_{k\in Z} a_k \mathbf X^k$


 * $\displaystyle g = \sum_{k\in Z} b_k \mathbf X^k$


 * $\displaystyle h = \sum_{k\in Z} c_k \mathbf X^k$

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

By Multiplication of Polynomials is Commutative, it is sufficient to prove that $\circ$ is left distributive over addition only.

Then

Therefore, $f \circ \left({ g + h }\right) = f \circ g + f \circ h$ for all polynomials $f$, $g$, $h$.

Therefore, polynomial multiplication is distributive over addition.