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Nowadays we can easily prove the following fact using polynomial long division:

If $a$ is a root of the polynomial $f$, then there exists a polynomial $g$ such that $f(x) = (x - a)g(x)$.

I can't imagine how to prove this without polynomial long division. I have two questions:

  1. Who invented polynomial long division, and when?
  2. Was the above theorem known before the discovery of polynomial long division? If so, how was it proven?
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  • $\begingroup$ Actually the first time I learned about this factorisation it was proved without the polynomial long division. You basically say that $f(x) = f(x) - f(a)$ and try to factorize that explicitly. $\endgroup$ – Maxime Lucas Apr 3 '15 at 15:55
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Up to notational conventions polynomial division algorithm was first described by an Arabic mathematician al-Samawal (1130-1180), who can also be credited with defining what we call polynomials today, see Who invented short and long division? In light of that the factorization theorem does not predate polynomial division.

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