Factoring Cubic, No Rational Roots

Question

I'm trying to find the values of x for which $x^3-3x^2+3x+1>0$, however, none of the roots from the rational roots test were factors. How would I go about solving it?

Answer 1

This can be written as $(x-1)^3+2>0$. Is that helpful?

Answer 2

The equation is equivalent to $$0<(x-1)^3+2$$ So $$1-\sqrt[3]{2}<x$$