13. The polynomial function f is defined by f(x) = x ^ 3 - 3x ^ 2 + 3x - 2 If 2 is one of the roots off, which of the following is also a root of f?
Answer: C
1/2 + ((√3/2)i) is also a root of f.
Since the polynomial f(x) has real coefficients, if a complex number is a root, then its conjugate must also be a root. Given that 2 is a root, we can utilize the properties of polynomials to find that 1/2 + ((√3/2)i) is indeed another root.
A) 1 + ((√5/2)i)
This option presents a complex number; however, it does not satisfy the conditions of being a root derived from the given polynomial. Its real part (1) does not align with the expected roots based on the polynomial's structure, and thus it cannot be a root of f.
B) 1 + ((√3/2)i)
Similar to option A, this complex number does not meet the necessary conditions to be a root of the polynomial. The real part does not correspond to the roots derived from the polynomial's behavior, ruling it out as a solution.
C) 1/2 + ((√3/2)i)
This option is valid as a root of the polynomial function f. The conjugate of this complex number, 1/2 - ((√3/2)i), would also be a root, satisfying the necessity for complex roots in polynomials with real coefficients.
D) 1/2 + (√3/2)
This option presents a real number, but it does not correspond to any root of the polynomial f. Substituting this value into the polynomial function yields a non-zero result, confirming it is not a root.
E) 1/2 + (√5/2)
Just like option D, this is a real number that does not satisfy the polynomial equation f(x) = 0. Therefore, it cannot be considered a root of the polynomial function.
Conclusion
The correct answer, 1/2 + ((√3/2)i), is a root of the polynomial function f, as it aligns with the requirement for complex roots when the polynomial has real coefficients. The other options either do not satisfy the polynomial equation or do not fit within the expected behavior of the roots given the context of the polynomial. Thus, option C is definitively the correct choice.