Answer
$(b),\ (c),$ and $(d)$
Work Step by Step
The domain of a rational expression is the whole set of real numbers, excluding the numbers that make the denominator equal to zero (since division of zero is not allowed).
Here, we exclude $x$ for which the denominator is equal to zero.
To find the numbers that will make the denominator zero, set the denominator equal to zero then solve the equation:
$ x^{3}-x=0\qquad$
Factor the binomial by factoring out $x$ to obtain:
$x(x^{2}-1)=0$
Factor the difference of two squares using the formula $a^2-b^2=(a-b)(a+b)$ to obtain:
$x(x-1)(x+1)=0$
Solve the equation using the Zero-Product Property by equating each factor to zero, then solve each equation to obtain:
\begin{align*}
x&=0 &\text{or}& &x-1=0& &\text{or}& &x+1=0\\
x&=0 &\text{or}& &x=1& &\text{or}& &x=-1
\end{align*}
Thus, the numbers that will be excluded from the domain are $0$, $-1$, and $1$..
Hence, the answer is $(b)$, $(c)$, and $(d)$.