3 roots of a 5th degree polynomial function f(x) are -2,2 and 4+i. which statement describes the number and nature of all roots for the function
f of x has 2 real roots and one imaginary root.
f of x has 3 real numbers and one imaginary root.
f of x has 5 real numbers and one imaginary root.
f of x has 3 real roots and 2 imaginary roots

Given that one of the roots is \(4 + i\), we must remember that the coefficients of polynomials with real numbers are real. This means that if a polynomial has a non-real (complex) root, its complex conjugate must also be a root. Therefore, the conjugate of \(4 + i\) is \(4 - i\), and it must also be a root.

Now, the polynomial is of degree 5, and we already know three of its roots: \(-2\), \(2\), and \(4 + i\). The conjugate \(4 - i\) will add another root, giving us 4 roots so far.

Since the degree of the polynomial is 5, there must be one more root. This root must be a real number because complex roots occur in conjugate pairs, and we only need one more root.

### Summary of the roots:
- 2 real roots: \(-2\), \(2\)
- 2 complex roots: \(4 + i\) and \(4 - i\)
- 1 remaining real root.

### Therefore, the statement that best describes the number and nature of all roots is:

**f of x has 3 real roots and 2 imaginary roots.**

Would you like more details on this explanation or have any further questions?

Here are 5 related questions you might consider:
1. What are the general properties of the roots of polynomials with real coefficients?
2. How can you factor a polynomial if you know its complex roots?
3. What is the significance of the degree of a polynomial function in relation to its roots?
4. Can a polynomial of degree 5 have all real roots?
5. How are the imaginary roots represented graphically on the complex plane?

**Tip:** Always remember that the roots of polynomials with real coefficients come in complex conjugate pairs if they are non-real.

3 roots of a 5th degree polynomial function f(x) are -2, 2 and 4+i. Which statement describes the number and nature of all roots for the function?
f of x has 2 real roots and one imaginary root.
f of x has 3 real numbers and one imaginary root.
f of x has 5 real numbers and one imaginary root.
f of x has 3 real roots and 2 imaginary roots.

Learn how to find the roots of a 5th-degree polynomial function where 3 roots are given: -2, 2, and 4+i. This solution explores the concept of complex conjugates and uses the Fundamental Theorem of Algebra to describe the total number and nature of the roots. Suitable for Grades 10-12.

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