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]

How would I solve, for example, (x2-x-6)/(x-1) without long division? (sorry, don't know how to use Latex)

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
$$\frac{x^2- x}{x- 1}+ \frac{-6}{x- 1}= \frac{x(x- 1)}{x- 1}+ \frac{-6}{x- 1}$$
$$= x+ \frac{-6}{x- 1}$$
so x- 1 divides into $x^2- 1$ x times with remainder -6.

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