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 three real roots and two imaginary roots.

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.

c) should be the answer.

qualitative observation deals with data that can be observed with our senses: sight, smell, touch, taste, and hearing. they do not involve measurements or numbers. for instance, colors, shapes, and textures of objects are all qualitative observations.