The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) roots in the complex number system, counting multiplicities. For a polynomial of degree 4, this means there are 4 roots in total.
Here are the possible outcomes when applying the Fundamental Theorem of Algebra to a polynomial equation of degree 4:
- Four real roots and no complex roots: This is possible if all roots are real.
- Three real roots and one complex root: This scenario is not possible, as complex roots must occur in conjugate pairs. If there is one complex root, there must be another, making the total number of real roots odd (not 3).
- Two real roots and two complex roots: This is possible. The two complex roots would form a conjugate pair.
- One real root and three complex roots: This scenario is not possible for the same reason as above; if there is one complex root, there must be another, meaning there must be an even number of complex roots.
- No real roots and four complex roots: This is also possible if all four roots are complex, which must include two conjugate pairs.
Based on the analysis, the valid outcomes are:
- Four real roots and no complex roots
- Two real roots and two complex roots
- No real roots and four complex roots
Therefore, the correct responses are:
- Four real roots and no complex roots
- Two real roots and two complex roots
- No real roots and four complex roots
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