The set of all polynomials. \[p=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}\]
over the commutative ring \(R\) (i.e. with \(a_{i}\in R,\,i=0,\ldots ,n,\,n\in \mathbb {N} \)), together with the component-wise addition and multiplication follow the usual rules of exponentiation \(X^{n}\cdot X^{m}:=X^{n+m}\) is called polynomial ring and denoted by \(R[X]\). Specifically, if \(p,q\in R[X]\) with
\[p=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{m}x^{m},\]
and
\[q=b_{0}+b_{1}x+b_{2}x^{2}+\cdots +b_{n}x^{n},\]
then the addition of the two polynomials in \(R[X]\) is defined as follows:
\[p+q:=r_{0}+r_{1}x+r_{2}x^{2}+\cdots +r_{k}x^{k},\quad\quad k = \max(m, n),\,r_{i}:=a_{i}+b_{i}\]
and
\[p\cdot q:=s_{0}+s_{1}x+s_{2}x^{2}+\cdots +s_{l}x^{l},\quad\quad l = m + n,\,s_{i}=a_{0}b_{i}+a_{1}b_{i-1}+\cdots +a_{i-1}b_{1}+a_{i}b_{0}.\]