Polynomial Addition
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 \(R\), with \(m \leq n\). We define polynomial addition over \(R\) to be the binary operation \(+\) given by the rule:
\[ +:(f(x),g(x)) \rightarrow f+g(x)=f(x)+g(x)=c_0+c_1x+\dots+c_nx^n \]where
\[ c_j=a_j+b_j \text{ and } a_j=0 \text{ for all } m < j \leq n \]