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?

Rationale

1/2 + ((√3/2)i) is also a root of f.

Given that the polynomial function f(x) is defined as f(x) = x^3 - 3x^2 + 3x - 2 and that 2 is a root, the polynomial can be factored. By using polynomial division or synthetic division, we can simplify f(x) to find the other roots, which include complex conjugates. The complex root 1/2 + ((√3/2)i) satisfies the condition of the polynomial.

A (1 + ((√5/2)i)) YOUR ANSWER

This choice does not satisfy the polynomial equation when substituted into f(x). The real part does not match the criteria established by the derived polynomial factors, and the imaginary part is not congruent with the complex conjugate requirement stemming from the presence of 2 as a root.

B (1 + ((√3/2)i)) YOUR ANSWER

This root fails to satisfy the polynomial function as well. Like the previous option, it does not fulfill the necessary conditions derived from the factorization of the cubic polynomial, leading to an incorrect evaluation of f(x).

C (1/2 + ((√3/2)i)) CORRECT

This option correctly represents one of the roots of the polynomial function f. When substituted back into the polynomial, it satisfies the equation, confirming it as a valid root alongside the known root of 2, due to the nature of complex roots appearing in conjugate pairs.

D (1/2 + (√3/2)) YOUR ANSWER

This choice is a real number and does not meet the criteria for a root of the polynomial. The presence of an imaginary component is essential because the polynomial has complex roots, and this option lacks that necessary characteristic.

E (1/2 + (√5/2)) YOUR ANSWER

Similar to option D, this choice is a real number and does not represent a root of the polynomial. It does not align with the expected roots based on the polynomial's nature and does not satisfy f(x) when evaluated.

Conclusion

The polynomial f(x) = x^3 - 3x^2 + 3x - 2 has 2 as one of its roots, indicating the presence of additional roots, including complex numbers. Among the given options, 1/2 + ((√3/2)i) is the only valid root, fulfilling the necessary conditions derived from the polynomial's structure. The other options either fail to satisfy the polynomial or do not conform to the expected nature of roots associated with complex conjugates.

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?

1/2 + ((√3/2)i) is also a root of f.

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