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.

Question

Gilbert Hodge

Mathematics

14 January, 17:11

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.

+2

Answers (1)

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Pinto · Answer 1

Because the polynomial is of the fifth degree, we expect five roots. If 4 + i is one of those roots, there must be a corresponding 4 - i. We can conclude that there are two complex roots and three real roots. The answer choice that involves the word "imaginary" cannot be correct; rather, we must use the word "complex" root to describe roots that have both real and imaginary parts. "f (x) has three real roots" is the correct answer here.