# Definition:Multiplication of Polynomials

## Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

Let $x \in R$.

Let:

$\ds f = \sum_{j \mathop = 0}^n a_j x^j$
$\ds g = \sum_{k \mathop = 0}^n b_k x^k$

be polynomials in $x$ over $S$ such that $a_n \ne 0$ and $b_m \ne 0$.

The product of $f$ and $g$ is defined as:

$\ds f g := \sum_{l \mathop = 0}^{m + n} c_l x^l$

where:

$\ds \forall l \in \set {0, 1, \ldots, m + n}: c_l = \sum_{\substack {j \mathop + k \mathop = l \\ j, k \mathop \in \Z} } a_j b_k$

### Polynomial Forms

Let $\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$ and $\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomial forms in the indeterminates $\set {X_j: j \in J}$ over $R$.

The product of $f$ and $g$ is defined as:

$\ds f \circ g := \sum_{k \mathop \in Z} c_k \mathbf X^k$

where:

$\ds c_k = \sum_{\substack {p + q \mathop = k \\ p, q \mathop \in Z} } a_p b_q$

### Polynomials as Sequences

Let:

$f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$

and:

$g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots}$

Then the operation of (polynomial) multiplication is defined as:

$f g := \tuple {c_0, c_1, c_2, \ldots}$

where $\ds c_i = \sum_{j \mathop + k \mathop = i} a_j b_k$