Question

Choose ,begin emphasis,all,end emphasis, the possible root combinations for a fifth degree polynomial.

Answer options with 6 options
A.
5 non-real roots

B.
1 real root and 4 nonreal roots

C.
2 real roots and 3 nonreal roots

D.
3 real roots and 2 nonreal roots

E.
4 real roots and 1 nonreal root

F.
5 real roots

Answer 1

A fifth degree polynomial can have various combinations of real and non-real roots, adhering to the Fundamental Theorem of Algebra and the nature of complex roots (which occur in conjugate pairs). Here are the permissible combinations:

1. **5 non-real roots** (not possible, since complex roots must occur in conjugate pairs)
2. **1 real root and 4 nonreal roots** (possible, as the remaining non-real roots can be two pairs of conjugate complex roots)
3. **2 real roots and 3 nonreal roots** (possible, which could be one pair of conjugate non-real roots and one single real root)
4. **3 real roots and 2 nonreal roots** (possible, with one pair of conjugate non-real roots)
5. **4 real roots and 1 nonreal root** (possible, with one non-real root that is counted as the remaining pair of conjugate complex roots)
6. **5 real roots** (possible, as all roots can be real)

The valid combinations for the roots of a fifth degree polynomial are thus:

- **B. 1 real root and 4 nonreal roots**
- **C. 2 real roots and 3 nonreal roots**
- **D. 3 real roots and 2 nonreal roots**
- **E. 4 real roots and 1 nonreal root**
- **F. 5 real roots**

So the correct options are:
- **B, C, D, E, F**

(Option A is not valid).