Choose all the possible root combinations for a fifth degree polynomial

The correct answers are:

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

Explanation: A fifth-degree polynomial can have a maximum of 5 roots, including complex (nonreal) roots. Since complex roots appear in conjugate pairs, the combinations of real and nonreal roots must follow these rules:

– If there are 5 roots, they can all be real (option F).

– If there are nonreal roots, they must appear in pairs, hence 1, 3, or 5 real roots are possible:

– 1 real root must be paired with 4 nonreal roots (option B).

– 2 real roots can pair with 3 nonreal roots (option C).

– 3 real roots can pair with 2 nonreal roots (option D).

– 4 real roots can pair with 1 nonreal root (option E).

Always remember that the total number of roots must equal the degree of the polynomial!