In this post, we dive into the inefficiency of multiplying polynomials using the traditional high school method, which has a complexity of $O(n^2)$. By connecting polynomials with the Fourier Transform and convolutions, we introduce a faster way to multiply polynomials with $O(nlogn)$ complexity. We explore how convolution in the time domain is equivalent to multiplication in the frequency domain and how converting polynomials to the frequency domain using the FFT can significantly speed up the process. By utilizing the FFT approach, polynomial multiplication becomes more efficient, challenging the conventional teaching methods. The comparison of time taken for multiplication using both naive and FFT methods further highlights the advantages of this innovative approach.
https://www.alvarorevuelta.com/posts/fft-polynomials