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

f(x) has three real roots and two imaginary roots.
The second imaginary root is the conjugate of the first. The third real root can be obtained by multiplying the two imaginary ones together.

Answer 2

Answer: The correct option is f(x) has three real roots and two imaginary roots.

Step-by-step explanation:

It is given that the roots of fifth degree polynomial function are -2, 2 and 4+i.

Since he degree of f(x) is 5, therefore there are 5 roots of the function either real or imaginary.

According to the complex conjugate root theorem, if a+ib is a root of a polynomial function f(x), then a-ib is also a root of the polynomial f(x).

Since 4+i is a root of f(x), so by complex conjugate rot theorem 4-i is also a root of f(x).

From the the given data the number of real roots is 2 and the number of 2. The number of complex roots is always an even number, so the last root must be a real number.

Therefore, the correct option is f(x) has three real roots and two imaginary roots.