Question

If 9i is a root of the polynomial function f(x), which of the following must also be a root of f(x)?

1) –9i
2) negative one-ninth
3) i
4) one-ninth
5) i 9 – i

Answer 1

The complex conjugate, −9i, must also be a root of the polynomial function f(x) if 9i is a root, according to the Complex Conjugate Root Theorem.

Step-by-step explanation:

If 9i is a root of the polynomial function f(x), then the complex conjugate, −9i, must also be a root of f(x). This is due to the Complex Conjugate Root Theorem which states that if a polynomial has real coefficients, then the non-real complex roots must occur in conjugate pairs. Therefore, the correct answer is −9i.

The other options provided do not necessarily represent roots that must accompany 9i for any polynomial with real coefficients. None of the other options are the conjugate of 9i. Also, it's important to know that powers and roots operate under specific rules for real numbers and do not automatically apply to imaginary numbers, so options involving reciprocals or fractional exponents are not directly related to the existence of the root 9i.