Factoring a large Diff of Squares
y = x^1000 - 1
I would like to factor out an x-1 from this but I don't remember which rules apply.
1) Do I apply something similar to a difference of squares and just keep going until I get (x-1)(....)?
(x^500)^2 - (1)^2 = (x^500+1)(x^500-1) = (x^500+1)(x^250+1)(x^250-1) = (x^500+1)(x^250+1)(x^125+1)(x^125-1) = now I'm stuck and I couldn't even get it to x-1 because I ran into an odd exponent ... well I'll try difference of cubes in that last factor...: = (x^500+1)(x^250+1)(x^125+1)(x^5-1)(x^10+x^5+1)... Okay, now I'm stuck, unless I remember how to factor (x^10+x^5+1)...
2) Or do I use some kind of Binomial Theorem.
Is it true that
x^1000 - 1 = (x-1)(x^999+x^998+x^997......+1) ?
3) Do I have to remember Pascal's Triangle at all?
For example, if number 2 above is true, then do I have to put in pascal's coefficients? ie.
(x-1)(1x^999 + 999x^998 + ...... + 1)?