# Definition:Multiplication of Polynomials

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## Definition

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

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

Let $x \in R$.

Let:

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

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:

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

where:

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

### Polynomial Forms

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

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

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

where:

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

### Polynomials as Sequences

Let:

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

and:

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

be polynomials over a field $F$.

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

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

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

## Also see

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain: Remarks $\text{(b)}$