Why does polynomial long division work?

BenB

So I'm in a college algebra class and I know how to do polynomial long division. I'm curious as to why polynomial long division works. I've looked at some proofs, but they use scary symbols that I don't understand (I am quite dumb). Do I need very high-level math to comprehend why polynomial long division works? What I'd like to see, if it's possible, is an example of a polynomial division problem being solved with just basic algebra. How would I solve, for example, (x²-x-6)/(x-1) without long division? (sorry, don't know how to use Latex)

jedishrfu

polynomial division is very similar to numerical long division.

A common form of polynomial long division is synthetic division:

http://en.wikipedia.org/wiki/Synthetic_division

which may show you how similar they are and why they work.

SteamKing

Have you tried factoring the numerator?

HallsofIvy

Since neither factor is x- 1, I don't believe factoring helps with the division.

Instead write this as
[tex]\frac{x^2- x}{x- 1}+ \frac{-6}{x- 1}= \frac{x(x- 1)}{x- 1}+ \frac{-6}{x- 1}[/tex]
[tex]= x+ \frac{-6}{x- 1}[/tex]
so x- 1 divides into [itex]x^2- 1[/itex] x times with remainder -6.

You could also use "synthetic division" as shown here: http://www.purplemath.com/modules/synthdiv.htm