Answer :
To determine which of the given options could be the complete list of the roots of a fourth-degree polynomial [tex]\( f(x) \)[/tex], we need to consider the properties of polynomials and their roots, especially regarding real and complex roots.
### Step-by-Step Analysis
1. Fundamental Theorem of Algebra:
- A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots, counting multiplicity.
2. Complex Conjugate Root Theorem:
- For polynomials with real coefficients, any complex roots must occur in conjugate pairs. So if [tex]\( a + bi \)[/tex] (where [tex]\( a, b \)[/tex] are real numbers and [tex]\( i \)[/tex] is the imaginary unit) is a root, then [tex]\( a - bi \)[/tex] must also be a root.
### Checking Each Option:
#### Option 1: [tex]\( 3, 4, 5, 6 \)[/tex]
- Analysis:
- All four roots are real numbers.
- This satisfies the requirement of having exactly four roots for a fourth-degree polynomial.
- There are no complex numbers, so we do not need to worry about conjugate pairs.
- Conclusion:
- This is a valid list of roots for a fourth-degree polynomial.
#### Option 2: [tex]\( 3, 4, 5, 6i \)[/tex]
- Analysis:
- Three roots are real numbers: [tex]\( 3, 4, 5 \)[/tex].
- One root is a purely imaginary number: [tex]\( 6i \)[/tex].
- For [tex]\( 6i \)[/tex] (a complex number) to be a root of a polynomial with real coefficients, [tex]\(-6i\)[/tex] must also be a root.
- Conclusion:
- This option is invalid as we do not have [tex]\(-6i\)[/tex] included in the list.
#### Option 3: [tex]\( 3, 4, 4 + i \sqrt{6}, 5 + \sqrt{6} \)[/tex]
- Analysis:
- Two roots are real numbers: [tex]\( 3, 4 \)[/tex].
- Two roots are complex numbers: [tex]\( 4 + i \sqrt{6} \)[/tex] and [tex]\( 5 + \sqrt{6} \)[/tex].
- For [tex]\( 4 + i \sqrt{6} \)[/tex] to be a root, the conjugate [tex]\( 4 - i \sqrt{6} \)[/tex] must also be a root, but it is not included.
- [tex]\( 5 + \sqrt{6} \)[/tex] is a real root, so it doesn't need a conjugate.
- Conclusion:
- This option is invalid as the complex root does not come with its conjugate pair.
#### Option 4: [tex]\( 3, 4, 5 + 1, -5 + i \)[/tex]
- Analysis:
- Two roots are real numbers: [tex]\( 3, 4 \)[/tex].
- One root is a complex number: [tex]\( -5 + i \)[/tex].
- For [tex]\( -5 + i \)[/tex] to be a root, its conjugate [tex]\( -5 - i \)[/tex] must also be a root, but it is not included.
- The term [tex]\( 5 + 1 \)[/tex] should read as 6, but it's a real number so it doesn't impact the conjugate pair rule.
- Conclusion:
- This option is invalid as it does not include the conjugate pair for the complex root [tex]\( -5 + i \)[/tex].
### Final Decision:
Based on the analysis, Option 1: [tex]\( 3, 4, 5, 6 \)[/tex] is the only valid complete list of roots for a fourth-degree polynomial.
Thus, the correct option is:
1. [tex]\( 3, 4, 5, 6 \)[/tex]
### Step-by-Step Analysis
1. Fundamental Theorem of Algebra:
- A polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots, counting multiplicity.
2. Complex Conjugate Root Theorem:
- For polynomials with real coefficients, any complex roots must occur in conjugate pairs. So if [tex]\( a + bi \)[/tex] (where [tex]\( a, b \)[/tex] are real numbers and [tex]\( i \)[/tex] is the imaginary unit) is a root, then [tex]\( a - bi \)[/tex] must also be a root.
### Checking Each Option:
#### Option 1: [tex]\( 3, 4, 5, 6 \)[/tex]
- Analysis:
- All four roots are real numbers.
- This satisfies the requirement of having exactly four roots for a fourth-degree polynomial.
- There are no complex numbers, so we do not need to worry about conjugate pairs.
- Conclusion:
- This is a valid list of roots for a fourth-degree polynomial.
#### Option 2: [tex]\( 3, 4, 5, 6i \)[/tex]
- Analysis:
- Three roots are real numbers: [tex]\( 3, 4, 5 \)[/tex].
- One root is a purely imaginary number: [tex]\( 6i \)[/tex].
- For [tex]\( 6i \)[/tex] (a complex number) to be a root of a polynomial with real coefficients, [tex]\(-6i\)[/tex] must also be a root.
- Conclusion:
- This option is invalid as we do not have [tex]\(-6i\)[/tex] included in the list.
#### Option 3: [tex]\( 3, 4, 4 + i \sqrt{6}, 5 + \sqrt{6} \)[/tex]
- Analysis:
- Two roots are real numbers: [tex]\( 3, 4 \)[/tex].
- Two roots are complex numbers: [tex]\( 4 + i \sqrt{6} \)[/tex] and [tex]\( 5 + \sqrt{6} \)[/tex].
- For [tex]\( 4 + i \sqrt{6} \)[/tex] to be a root, the conjugate [tex]\( 4 - i \sqrt{6} \)[/tex] must also be a root, but it is not included.
- [tex]\( 5 + \sqrt{6} \)[/tex] is a real root, so it doesn't need a conjugate.
- Conclusion:
- This option is invalid as the complex root does not come with its conjugate pair.
#### Option 4: [tex]\( 3, 4, 5 + 1, -5 + i \)[/tex]
- Analysis:
- Two roots are real numbers: [tex]\( 3, 4 \)[/tex].
- One root is a complex number: [tex]\( -5 + i \)[/tex].
- For [tex]\( -5 + i \)[/tex] to be a root, its conjugate [tex]\( -5 - i \)[/tex] must also be a root, but it is not included.
- The term [tex]\( 5 + 1 \)[/tex] should read as 6, but it's a real number so it doesn't impact the conjugate pair rule.
- Conclusion:
- This option is invalid as it does not include the conjugate pair for the complex root [tex]\( -5 + i \)[/tex].
### Final Decision:
Based on the analysis, Option 1: [tex]\( 3, 4, 5, 6 \)[/tex] is the only valid complete list of roots for a fourth-degree polynomial.
Thus, the correct option is:
1. [tex]\( 3, 4, 5, 6 \)[/tex]
