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 $3$?
$\begingroup$
$\endgroup$
5
-
2$\begingroup$ "Highest term" means highest degree term. Polynomial long division is analogous to long division of integers in radix representation. $\endgroup$– Bill DubuqueFeb 10, 2011 at 0:05
-
$\begingroup$ Also: the $x$ in a polynomial does not represent an unknown number. $\endgroup$– Arturo MagidinFeb 10, 2011 at 4:45
-
1$\begingroup$ 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 DubuqueFeb 10, 2011 at 18:51
-
1$\begingroup$ Maybe we could improve the wikipedia article on Polynomial Long Division then? I wonder if others have been similarly confused. $\endgroup$– David KohlerMar 31, 2011 at 17:03
-
$\begingroup$ @David Kohler: +1 great idea. I did little bit, but it can still be improved. $\endgroup$– The StudentApr 1, 2011 at 12:45
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
"highest" here means not "greatest", but the one with the highest power of $x$ (which in this case is $x$).
-
$\begingroup$ that is, the "highest-degree" term $\endgroup$ Apr 10, 2011 at 7:22