1. **State the problem:** Find the rational zeros of the polynomial $$f(x) = x^3 - 5x^2 + 2x + 1$$ using the Rational Zero Theorem. 2. **Recall the Rational Zero Theorem:** Possible rational zeros are of the form $$\pm \frac{p}{q}$$ where $p$ divides the constant term and $q$ divides the leading coefficient. 3. **Identify $p$ and $q$:** - Constant term = 1, so possible $p = \pm 1$ - Leading coefficient = 1, so possible $q = \pm 1$ 4. **List possible rational zeros:** $$\pm 1$$ 5. **Test $x=1$:** $$f(1) = 1^3 - 5(1)^2 + 2(1) + 1 = 1 - 5 + 2 + 1 = -1 \neq 0$$ 6. **Test $x=-1$:** $$f(-1) = (-1)^3 - 5(-1)^2 + 2(-1) + 1 = -1 - 5 - 2 + 1 = -7 \neq 0$$ 7. Since neither $1$ nor $-1$ is a zero, there are no rational zeros. **Final answer:** The polynomial $$f(x) = x^3 - 5x^2 + 2x + 1$$ has no rational zeros according to the Rational Zero Theorem.