Multiplication of Polynomials Distributes over Addition

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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

\(\displaystyle f \circ \paren {g + h}\) \(=\) \(\displaystyle \sum_{k \mathop \in Z} \sum_{p \mathop + q \mathop = k} a_p \paren {b_q + c_q} \mathbf X^k\) by the definitions of polynomial multiplication and addition
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in Z} \sum_{p \mathop + q \mathop = k} \paren {a_p b_q + a_p c_q} \mathbf X^k\) by the properties of finite sums
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in Z} \sum_{p \mathop + q \mathop = k} a_p b_q \mathbf X^k + \sum_{k \mathop \in Z} \sum_{p + q = k} a_p c_q \mathbf X^k\) by the properties of finite sums
\(\displaystyle \) \(=\) \(\displaystyle f \circ g + f \circ h\) by the definitions of polynomial multiplication and addition

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.

$\blacksquare$