Q.Find all values of $x$ for which $\dfrac{x^3(x-1)}{(x-2)}>0$.

To solve a rational inequality $\dfrac{f(x)}{g(x)}\gtrless0$ (or with a constant on the other side, first move it to one side and combine into a single fraction): factor $f$ and $g$ completely, mark every critical point -- the zeros of $f$ (where the expression equals $0$) and the zeros of $g$ (where it is undefined) -- on the number line, and build a sign table: for each resulting interval, record the sign of every individual linear (or irreducible-quadratic, always positive) factor, and multiply/divide them to get the overall sign of the expression on that interval.

Inclusion rule. A zero of the numerator is included in the solution set only when the inequality is non-strict ($\le$ or $\ge$); a zero of the denominator is always excluded, even for a non-strict inequality, since the expression is undefined there.

Odd vs. even multiplicity. A factor raised to an odd power (like $x^3$) still changes the overall sign at that root, exactly like a simple linear factor; a factor raised to an even power never changes sign (it stays $\ge0$) and can usually be dropped from the sign analysis (except to note where it makes the expression exactly $0$).