## College Algebra (10th Edition)

$(c)$
RECALL: The denominator of a rational expression is not allowed to be equal to zero since division of zero is undefined. Factor the denominator to obtain: $\dfrac{-9x^2-x+1}{x^3+x} = \dfrac{-9x^2-x+1}{x(x^2+1)}$ 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: $\begin{array}{ccc} \\&x = 0 &\text{ or } &x^2+1 = 0 \\&x=0 &\text{ or } &x^2=-1 \\\end{array}$ No real number satisfies the second equation. Thus, in the given expression, $x$ cannot be equal to $0$ since it will make the denominator equal to 0. Therefore, the answer is: $(c)$.