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.

answer: the primary difference between a hypothesis and a theory is that b hypothesis is falsifiable, but a theory has been proven to be correct. theory is an explanation of something that is happening but hypothesis is a guess of something that can happen

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