From what I've read about it all in the past, I feel like my intelligence level sits precisely between DCT and DFT. I seem to be able to mostly get Discrete Cosine Transform, but when confronted with Discrete Fourier Transform my brain simply shuts off and melts.
If you wanted to multiply two polynomials f and g of degree n, you could do distribution and the do n squared multiplications. Instead, evaluate the two separate polynomials at 2n special points, multiply the results pointwise, and then interpolate. The magical part is there are some specific points which speed up the evaluation and interpolation steps using a divide and conquer type algorithm.