How would you factor y^2 + y - 2? Now let x^6 = y.
I'll give it a try:
The factors of 2 are: 1 ; 2 and -1 ; -2 if I combine -1 and 2, the factors of the given quadratic are: (y - 1)(y + 2) expanding this gives: y² + y - 2 ; subst. y = x^6 gives: (x^6 - 1) (x^6 + 2).
To factor anything further out: ( x^6 + 2) won't factor into anything of the first degree, (just leaving it in this form).
Now : (x^6 - 1) ; I'll first check if 1 and -1 are roots of this expression: (x = 1) ; (1^6) - 1 = 0 so 1 is a root; ( x = -1) ; (-1^6) - 1 = 0, so -1 is also a root. Multiply these (x + 1) (x - 1) = (x^2 - 1).
I'll divide (x^6 -1 ) by (x^2 - 1) the result is: x^4+ x^2 + 1 this factors in to (x² + x + 1) (x² - x + 1).
So: x^12 + x^6 - 2 = (x + 1) (x - 1) (x² + x + 1) (x² - x + 1) (x^6 +2).
Thanks.
Luke.