1. The problem asks for the possible rational roots of the polynomial $$f(x) = 3x^4 - x^3 + x^2 - x + 5$$. 2. We use the Rational Root Theorem, which states that any possible rational root, expressed as $$\frac{p}{q}$$, must have $$p$$ as a factor of the constant term and $$q$$ as a factor of the leading coefficient. 3. Here, the constant term is 5, and the leading coefficient is 3. 4. Factors of 5 (constant term) are $$\pm 1, \pm 5$$. 5. Factors of 3 (leading coefficient) are $$\pm 1, \pm 3$$. 6. Therefore, the possible rational roots are all combinations $$\pm \frac{p}{q}$$ where $$p$$ divides 5 and $$q$$ divides 3: $$\pm 1, \pm \frac{1}{3}, \pm 5, \pm \frac{5}{3}$$ 7. This matches option c. Final answer: {\pm 1, \pm \frac{1}{3}, \pm 5, \pm \frac{5}{3}}