To determine the nature of the roots of the given third degree polynomial, we start with the given information that one of its roots is [tex]\(-5 + 2i\)[/tex].
In mathematics, polynomial equations with real coefficients that have complex roots must have those complex roots come in conjugate pairs. Therefore, the complex conjugate of [tex]\(-5 + 2i\)[/tex] is [tex]\(-5 - 2i\)[/tex]. This indicates that if [tex]\(-5 + 2i\)[/tex] is a root, then [tex]\(-5 - 2i\)[/tex] must also be a root.
Given the polynomial is of the third degree, it must have exactly three roots in total. We have already identified two roots: [tex]\(-5 + 2i\)[/tex] and [tex]\(-5 - 2i\)[/tex].
Since there are exactly three roots for this polynomial and we already accounted for two of them, the third root must be a real number. This is because adding another imaginary or complex root would either exceed the total number of roots for a third degree polynomial or not satisfy the requirement of having real coefficients.
Here's a summary of the roots:
1. [tex]\(-5 + 2i\)[/tex] (imaginary root)
2. [tex]\(-5 - 2i\)[/tex] (imaginary root)
3. One real root
Thus, the statement that best describes the number and nature of all roots of this polynomial function is:
[tex]\[ f(x) \text{ has two imaginary roots and one real root.} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{f(x) \text{ has two imaginary roots and one real root.}} \][/tex]
