Multiplication of Polynomials Distributes over Addition

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

Proof
Let $\struct {R, +, \circ}$ be a commutative ring with unity.

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

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

Let


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


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


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

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

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

Then

Therefore, $f \circ \paren {g + h} = f \circ g + f \circ h$ for all polynomials $f, g, h$.

Therefore, polynomial multiplication is distributive over addition.