Fast Fourier Transform, Algorithms, Pattern Matching, Convolution, Combination Sum.
a_j*b_(i-j) for all j ranging from 0 to n-1: I would request the reader to verify and convince themselves the coefficients by multiplying the 2 polynomials by hand. Thus multiplying 2 integers can also be represented as above, i.e. multiplying 2 polynomials and then evaluating the final polynomial