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?

• $f(x)$ has three real roots.
• $f(x)$ has two real roots and one imaginary root.
• $f(x)$ has five real roots.
• $f(x)$ has three real roots and two imaginary roots.

Explanation:

Step1: Recall Complex Conjugate Root Theorem

For a polynomial with real coefficients, if \(a + bi\) is a root, then its complex conjugate \(a - bi\) is also a root. Here, the polynomial is a fifth - degree polynomial with real coefficients (implied, as we are dealing with roots of a polynomial function in the real - valued function context), and one of the roots is \(4 + i\). So, by the complex conjugate root theorem, \(4 - i\) must also be a root.

Step2: Count the number of real and imaginary roots

We know three roots initially: \(-2\) (real), \(2\) (real), and \(4 + i\) (imaginary). Then, from the complex conjugate root theorem, \(4 - i\) (imaginary) is also a root. Now, since it is a fifth - degree polynomial, the total number of roots (counting multiplicities) is 5. We already have 4 roots: \(-2\), \(2\), \(4 + i\), \(4 - i\). So the fifth root must be a real number (because if it were imaginary, its conjugate would also have to be a root, which would make the degree at least 6, but the polynomial is of degree 5).

So the real roots are \(-2\), \(2\), and the fifth root (total of 3 real roots), and the imaginary roots are \(4 + i\) and \(4 - i\) (total of 2 imaginary roots).

Answer:

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