Question

What are the possible outcomes when you apply the Fundamental Theorem of Algebra to a polynomial equation of degree 4?

(Choose all that apply.)

(2 points)
Responses

Four real roots and no complex roots
Four real roots and no complex roots

Three real roots and one complex root
Three real roots and one complex root

Two real roots and two complex roots
Two real roots and two complex roots

One real roots and three complex roots
One real roots and three complex roots

No real roots and four complex roots
No real roots and four complex roots

Answer 1

Select:
- Four real roots and no complex roots
- Two real roots and two complex roots
- No real roots and four complex roots

Reason: A degree-4 polynomial has exactly 4 roots in C (counting multiplicity). If coefficients are real, nonreal roots come in conjugate pairs, so the number of nonreal roots must be 0, 2, or 4 (giving 4, 2, or 0 real roots).