Answer the following questions about the equation below. \( x^{3}-17 x-4=0 \) (a) List all rational roots that are possible according to the Rational Zero Theorem. Choose the correct answer below. A. \( \pm 1 \) B. \( \pm 4 \) C. \( -1,-2,-4 \) D. \( \pm 1, \pm 2, \pm 4 \)

The Rational Root Theorem states that any potential rational root of a polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 \), is in the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (in this case, -4) and \( q \) is a factor of the leading coefficient (which is 1 here). The factors of -4 are \( \pm 1, \pm 2, \pm 4 \) and since the leading coefficient is 1, the possible rational roots are just those factors. So, the correct answer is D: \( \pm 1, \pm 2, \pm 4 \). Now that we’ve covered that, remember to check each candidate root by substituting back into the original equation to see if they yield zero. It can be quite the adventure to find the right root, but it’s all part of the math journey!