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?.

Answer 1

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

Given

Three roots of a fifth-degree polynomial function f(x) are –2, 2, and 4 i.

Complex conjugate

The complex conjugate of a complex number is the number with an equally real part and an imaginary part equal in magnitude but opposite in sign.

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

Since the 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 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.

To know more about imaginary roots click the link given below.

brainly.com/question/1375079