Factoring by grouping with $3$ terms only?

Question

I was wondering how can I factor this equation: $x^3-2x+1$ by grouping, I have this doubt because the solution is $(x-1)(x^2+x-1)$ and this seems clearly obtain by factoring by grouping but I have always thought that this method was possible only for 4 terms equations. So I tried to do my best and I factored it in this way: $$(x^3-2x)+1 \\ x(x^2-2)+1 $$ But I'm stuck here and I can't find the way to get to the right solution, I would really appreciate your attention and a clarification of this doubt of my mine. Thanks for your attention and have a good day!

Answer 1

we have $$x^3-2x+1=0$$ one solution is $x=1$ since $1^3-2+1=0$ and you can divide your equation by $x-1$, and we get $$x^3-2x+1=\left( x-1 \right) \left( {x}^{2}+x-1 \right)$$

Answer 2

While Dr. Graubners answer is correct, to get the result by factoring by grouping you can do the following:

$x^3 - 2x + 1$

$x^3 - x - x + 1$

$(x^3 - x) - (x - 1)$

$x(x^2 - 1) - (x - 1)$ (Notice the difference of squares)

$x(x - 1)(x + 1) - (x - 1)$

Now you can factor out your $(x-1)$, combine the leftovers, and get the solution you were given.