Polynomial Multiplication
Let \(R\) be a ring, and let \(f(x)=a_0+a_1x+\dots+a_mx^m\) and \(g(x)=b_0+b_1x+\dots+b_nx^n\) polynomials over \(R\). We define _polynomial multiplication over \(R\) to be the binary operation \(\cdot\) given by the rule:
\[ \cdot:(f(x),g(x)) \rightarrow fg(x)=f(x)g(x)=c_0+c_1x+\dots+c_kx^k \]where
\[ c_j=\sum_{i=0}^j{a_ib_{j-1}} \text{ and } k=m+n \]