Answer
$(b)$, $(c)$, and $(d)$
Work Step by Step
RECALL:
The denominator of a rational expression is not allowed to be equal to zero since division of zero is undefined.
Factor the denominator using the formula $a^2-b^2=(a-b)(a+b)$ to obtain:
$\dfrac{x^2+5x-10}{x^3-x} = \dfrac{x^2+5x-10}{x(x^2-1)}=\dfrac{x^2+5x-10}{x(x-1)(x+1)}$
Find the values of $x$ that will make the denominator equal to zero by using the zero-factor theorem.
Equate each factor of the denominator to zero then solve each equation to obtain:
$\begin{array}{ccccc}
\\&x=0 &\text{ or } &x-1 = 0 &\text{ or } &x + 1 = 0
\\&x=0 &\text{ or } &x = 1 &\text{ or } &x=-1
\\\end{array}$
Thus, in the given expression, $x$ cannot be equal to $-1, 0$ or $1$ since they make the denominator equal to 0.
Therefore, the answer is: $(b)$, $(c)$, and $(d)$.
