Question

Three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. Which statement describes the number and nature of all roots for this function?
f(x) has two real roots and one imaginary root.
f(x) has three real roots.
f(x) has five real roots.
f(x) has three real roots and two imaginary roots.

Answer 1

Correct answer: "f(x) has three real roots and two imaginary roots."

Reason: If the polynomial has real coefficients, nonreal complex roots occur in conjugate pairs, so 4 + i implies 4 − i is also a root. That gives two nonreal roots (4 ± i) and the two given real roots (−2 and 2). A 5th-degree polynomial must have one more root; since nonreal roots come in pairs, the remaining root must be real. Hence three real roots and two imaginary (nonreal) roots.