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? a. f(x) has two real roots and one imaginary root. b. f(x) has three real roots. c. f(x) has five real roots. d. f(x) has three real roots and two imaginary roots.

Step-by-step

Given:

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?

A. f(x) has two real roots and one imaginary root.

B. f(x) has three real roots.

C. f(x) has five real roots.

D. f(x) has three real roots and two imaginary roots.

Solution:

We know from the fundamental theorem of algebra that every non-constant single variable polynomial with real coefficients of degree N has exactly N roots, multiplicities included.

Therefore there are five roots of in polynomial function f(x).

Of the five roots, we are already given that 2 of them are real, one of them is complex.

Since for a polynomial with real coefficients, each complex root (4+i), has its conjugate (4-i) as another root, so there is a minimum of 2 real and 2 complex roots.

The remaining (fifth) root must therefore be real, since we cannot have an odd number of complex roots.  This gives a total of three real and two complex roots.

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