Definition:Multiplication of Polynomials

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({S, +, \circ}\right)$ 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:
 * $\displaystyle \forall l \in \left\{{0, 1, \ldots, m+n}\right\}: c_l = \sum_{\substack{j \mathop + k \mathop = l \\ j, k \mathop \in \Z}} a_j b_k$

Also see

 * Product of Polynomials is Polynomial