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 conjugate root theorem

For polynomials with real coefficients (the standard assumption unless stated otherwise), complex roots and irrational roots with radicals come in conjugate pairs. A fourth-degree polynomial has exactly 4 roots (counting multiplicities).

Step2: Analyze each option

• Option 1: $3, 4, 5, 6$: These are all real roots, 4 total, which is valid for a 4th-degree polynomial.
• Option 2: $3, 4, 5, 8i$: $8i$ has no conjugate $-8i$, so this cannot be a complete list for a real-coefficient polynomial.
• Option 3: $3, 4, 4+i\sqrt{6}, 5+\sqrt{6}$: $4+i\sqrt{6}$ lacks its conjugate $4-i\sqrt{6}$, and $5+\sqrt{6}$ lacks its conjugate $5-\sqrt{6}$, so this is invalid.
• Option 4: $3, 4, 5+i, -5+i$: $5+i$ lacks its conjugate $5-i$, and $-5+i$ lacks its conjugate $-5-i$, so this is invalid.

Answer:

3, 4, 5, 6