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The wikipedia example of Polynomial Long Division starts with:

Divide the first term of the numerator by the highest term of the denominator

if the denominator is $x-3$, and we don't know $x$, how to know which is highest? $x$ or $3$?

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    $\begingroup$ "Highest term" means highest degree term. Polynomial long division is analogous to long division of integers in radix representation. $\endgroup$ – Bill Dubuque Feb 10 '11 at 0:05
  • $\begingroup$ Also: the $x$ in a polynomial does not represent an unknown number. $\endgroup$ – Arturo Magidin Feb 10 '11 at 4:45
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    $\begingroup$ Arturo: But in fact $\rm\:x\:$ does represent an "unknown number". More precisely it represents a universal (generic) $\rm\:R$-algebra element. $\rm\ R[x]$ is defined so that the equations true in it are precisely those ring-theoretic equations that hold true in every $\rm\ R$-algebra. Therefore an equation is true in $\rm\:R[x]\:$ iff it is a universal identity of $\rm\:R$-algebras. For example see my post here and see here and see here. $\endgroup$ – Bill Dubuque Feb 10 '11 at 18:51
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    $\begingroup$ Maybe we could improve the wikipedia article on Polynomial Long Division then? I wonder if others have been similarly confused. $\endgroup$ – David Kohler Mar 31 '11 at 17:03
  • $\begingroup$ @David Kohler: +1 great idea. I did little bit, but it can still be improved. $\endgroup$ – The Student Apr 1 '11 at 12:45
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"highest" here means not "greatest", but the one with the highest power of $x$ (which in this case is $x$).

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  • $\begingroup$ that is, the "highest-degree" term $\endgroup$ – James Edward Lewis II Apr 10 '11 at 7:22

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