I suppose you could use

Descartes' rule of signs to find an approximation for the

*number* of roots.

But you are

__really__ asking about the

rational roots test, I think, which is a small expansion of the factor theorem.

You have to consider that some factors might be of the form $\displaystyle (3x\pm k)$, where 'k' refers to any of the factors of $\displaystyle 2$, by testing $\displaystyle f(\frac{-k}{3})$, as you would with the factor theorem. You should find that there are indeed more factors to this polynomial.