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 two roots of this functions are the exact roots and one is an imaginary root. The -2 is the root of a negative number -16 because if you compute the -2 on its 3rd power you come up -16, the 2 is the root of 16 and the 4+i contains a imaginary number that is why it is an imaginary root

Answer 2

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

Step-by-step explanation:

It is given that the three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. In which -2,2 are real root and 4+i is imaginary root.

The number of roots of a polynomial is equal to the degree of that polynomial.

Since the degree of polynomial function is 5, therefore the function has total 5 roots.

According to the complex conjugate root theorem, if a+ib is a root of a polynomial, then its conjugate a-ib is also a root of that function. The number of imaginary roots are always an even number.

Since 4+i is a root of the polynomial, therefore 4-i is also a root of the function.

2 roots are real and 2 roots are imaginary. The remaining 1 root must be real because the number of imaginary roots can not be odd.

Therefore the function f(x) has three real roots and two imaginary roots.