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

Also see

 * Product of Polynomials is Polynomial