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Multiplication needs to be automatic for doing algebra: FOIL and factoring polynomials immediately come to mind.

Long division, at least conceptually, is also present in algebra and partial fractions.

Indeed. And once you can mechanically do either of these and understand the process, you can defer the mechanics to a machine because continued mastery of a heuristic like FOIL is pretty irrelevant.

Similarly, algebra needs to be automatic to do calculus.

Everything builds on everything else.

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