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Haskell, 45 bytes foldl(\a b->zipWith((-).(*b))(0:a)$a++[0])[1] Try it online! Uses the "highest power first" convention. Sometimes multiplies the polynomial by $(-1)$.
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#1: Initial revision
# [Haskell], 45 bytes <!-- language-all: lang-hs --> foldl(\a b->zipWith((-).(*b))(0:a)$a++[0])[1] [Try it online!][TIO-ktfe3v9c] [Haskell]: https://www.haskell.org/ [TIO-ktfe3v9c]: https://tio.run/##NYyxDsIgFAB3v@INDmALqbo1KYuuTg4OlOG1hZZIKSm4@PFiE@NwudxyE8andi6bps1mcYMjLULHxNuGh00TIYxycugoJVWNdI9FIStF5VHlGa2HBkadLotP2qcIQjQwY7gBCav1iRtOVo1DXd/T1iMT8rq8OqcVpdxZr2OWare9Nko4/fSvEs7q0xuHY8ysD@EL "Haskell – Try It Online" Uses the "highest power first" convention. Sometimes multiplies the polynomial by $(-1)$.