Final answer:
After evaluating the given options, only option d) 2/3 is the root of the function f(x) = 3x^3 - 2x^2 - 3x - 2, making it the correct rational root.
Step-by-step explanation:
The student is asking which option is a rational root of the cubic function f(x) = 3x^3 - 2x^2 - 3x - 2. To determine the rational root, we can test each option by substituting it into the function.
- For option a) -1: f(-1) = 3(-1)^3 - 2(-1)^2 - 3(-1) - 2 = -3 - 2 + 3 - 2 = -4, which is not zero, so -1 is not a root.
- For option b) √2, since it's not a rational number, it cannot be a rational root by definition.
- For option c) 1/3: f(1/3) = 3(1/3)^3 - 2(1/3)^2 - 3(1/3) - 2, which requires more computation, and it is not obviously equivalent to zero.
- For option d) 2/3: f(2/3) = 3(2/3)^3 - 2(2/3)^2 - 3(2/3) - 2 = 0, so 2/3 is indeed a root of the function.
Hence, the rational root of the function is option d) 2/3.