Question

How do you factor `x^3 - 2x + 1`?

Answer 1

It is

`x^3-2x+1=x^3-x-x+1=x(x^2-1)-(x-1)=x(x-1)(x+1)-(x-1)=(x-1)*(x^2+x-1)`

Now for the trinomial `x^2+x-1` the roots are

`x_(1,2)=[-1^2(+-)sqrt(1^2+4)]/2=>x_1=(-1+sqrt5)/2 and x_2=(1+sqrt5)/2`

So finally the factorization is

`x^3-2x+1=-1/4 (x-1)*(-2 x+sqrt(5)-1)*(2 x+sqrt(5)+1)`