This always become a problem for me when there is a fraction in an equation, everything just tends to be harder. I multiplied by $3$ on both sides to get rid of $\frac{1}{3}$ making the equation become $3y = x^3-6x^2-15x+6$ and I tried using "synthetic division" but I couldn't get the remainder to become $0$. Is there any simpler way to figure this out? I am just trying to find $x$-axis intercept point, where $y = 0$.
The problem is in the image
Well, here is a plot of the equation:
Looking at a plot and considering the rational root theorem, I do not believe the equation has any rational roots. Have you thought about using some numerical analysis technique such as the bisection method?