factorising hard functionWatch
So when you factorise out x^-2 you get x^-2(x^4-2x^2+1) which factorise to x^-2(x^2-1)(x^2-1). Now multiply x^-1 into each of the brackets, and remove the x^-2 in front, giving (x-x^-1)^2
So this question, the mark scheme factorises to show that f'(x) is always greater than 0 (but that's irrelevant). What I'm trying to figure out is how to factorise such an expression in the first place. I understand that the brackets multiply out to become f'(x), but if I were to come across such a function in an exam I would be at a loss on how to even begin to factorise it into said bracket. It just... doesn't look like it should factorise. Does anyone have any insight or train of thought that can help me understand this more clearly / make it easier to see how to factorise this function, because from my POV the mark scheme basically just pulls it out of thin air.
So to factorise x^2 -2x^0 + x^-2 I formed 2 brackets with (x+?)(x+?). Then I dealt with the x^-2 term as the second part of each bracket giving (x-x^-1)(x-x^-1). Then I checked that the middle term could be found from these 2 brackets