1. **State the problem:** We are given the polynomial function $$f(x) = 2x^3 + x^2 - 18x - 9$$ and need to determine which of the statements A, B, C, or D is true. 2. **Recall the factor theorem:** If $$x - a$$ is a factor of $$f(x)$$, then $$f(a) = 0$$. 3. **Check statement A: "x - 3 is a factor of f(x)".** Calculate $$f(3)$$: $$f(3) = 2(3)^3 + (3)^2 - 18(3) - 9 = 2(27) + 9 - 54 - 9 = 54 + 9 - 54 - 9 = 0$$ Since $$f(3) = 0$$, by the factor theorem, $$x - 3$$ is a factor of $$f(x)$$. 4. **Check statement B: "f(3) \neq f(-1/2)".** Calculate $$f(-1/2)$$: $$f(-\frac{1}{2}) = 2\left(-\frac{1}{2}\right)^3 + \left(-\frac{1}{2}\right)^2 - 18\left(-\frac{1}{2}\right) - 9 = 2\left(-\frac{1}{8}\right) + \frac{1}{4} + 9 - 9 = -\frac{1}{4} + \frac{1}{4} + 9 - 9 = 0$$ Calculate $$f(3)$$ again (already found to be 0). Since $$f(3) = f(-\frac{1}{2}) = 0$$, statement B is false. 5. **Check statement C: "2x - 1 is a factor of f(x)".** If $$2x - 1$$ is a factor, then $$f\left(\frac{1}{2}\right) = 0$$. From step 4, $$f\left(\frac{1}{2}\right)$$ is: $$f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^2 - 18\left(\frac{1}{2}\right) - 9 = 2\left(\frac{1}{8}\right) + \frac{1}{4} - 9 - 9 = \frac{1}{4} + \frac{1}{4} - 9 - 9 = \frac{1}{2} - 18 = -\frac{35}{2} \neq 0$$ So statement C is false. 6. **Check statement D: "f(1/2) = 0".** From step 5, $$f\left(\frac{1}{2}\right) = -\frac{35}{2} \neq 0$$, so statement D is false. **Final answer:** Statement A is true.