Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. We note first that the new coefficients are given as
\begin{split} \delta_0=&\alpha_0\beta_0\\ \delta_1=&\alpha_1\beta_0+\alpha_0\beta_1\\ \delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ \delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ \delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ \delta_5=&\alpha_2\beta_3.\\ \end{split}We note that \alpha_i=0 except for i\in \left\{0,1,2\right\} and \beta_i=0 except for i\in\left\{0,1,2,3\right\} .
We can then rewrite the coefficients \delta_j using a discrete convolution as
\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j,or as a double sum with restriction l=i+j
\delta_l = \sum_{ij}\alpha_i\beta_{j}.