Multiplication of Polynomials Distributes over Addition

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Theorem

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

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

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

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.

$\blacksquare$

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