Question

the polynomial function f(x) is a fourth degree polynomial. which of the following could be the complete list of the roots of f(x)?
○ 3, 4, 5, 6
○ 3, 4, 5, 6i
○ 3, 4, 4+i√6, 5+√6
○ 3, 4, 5+i, −5+i

Explanation:

Step1: Recall Polynomial Root Rules

For a polynomial with real coefficients (implied here as standard), complex roots (with non - real imaginary parts) come in conjugate pairs, and irrational roots (of the form \(a+\sqrt{b}\) where \(b\) is not a perfect square) come in conjugate pairs (\(a - \sqrt{b}\)). A fourth - degree polynomial has 4 roots (counting multiplicities).

Step2: Analyze Option 1: 3, 4, 5, 6

These are all real roots. A fourth - degree polynomial can have 4 real roots. Let's check other options to be sure.

Step3: Analyze Option 2: 3, 4, 5, 6i

The complex root \(6i\) has a conjugate \(- 6i\) which is not in the list. So this cannot be the complete list of roots for a polynomial with real coefficients (since if the polynomial has real coefficients, complex roots must come in conjugate pairs).

Step4: Analyze Option 3: \(3,4,4 + i\sqrt{6},5+\sqrt{6}\)

The complex root \(4 + i\sqrt{6}\) has a conjugate \(4 - i\sqrt{6}\) which is not in the list. The irrational root \(5+\sqrt{6}\) has a conjugate \(5 - \sqrt{6}\) which is not in the list. So this cannot be the complete list of roots.

Step5: Analyze Option 4: \(3,4,5 + i,-5 + i\)

The complex root \(5 + i\) has a conjugate \(5 - i\) which is not in the list. The complex root \(-5 + i\) has a conjugate \(-5 - i\) which is not in the list. So this cannot be the complete list of roots.

Answer:

A. 3, 4, 5, 6