# Polynomial Long Division explanation

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 $3$?

• "Highest term" means highest degree term. Polynomial long division is analogous to long division of integers in radix representation. Feb 10, 2011 at 0:05
• Also: the $x$ in a polynomial does not represent an unknown number. Feb 10, 2011 at 4:45
• 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. Feb 10, 2011 at 18:51
• Maybe we could improve the wikipedia article on Polynomial Long Division then? I wonder if others have been similarly confused. Mar 31, 2011 at 17:03
• @David Kohler: +1 great idea. I did little bit, but it can still be improved. Apr 1, 2011 at 12:45

"highest" here means not "greatest", but the one with the highest power of $x$ (which in this case is $x$).