One root of a third-degree polynomial function [tex]\( f(x) \)[/tex] is [tex]\(-5 + 2i\)[/tex]. Which statement describes the number and nature of all roots for this function?

A. [tex]\( f(x) \)[/tex] has two real roots and one imaginary root.
B. [tex]\( f(x) \)[/tex] has two imaginary roots and one real root.
C. [tex]\( f(x) \)[/tex] has three imaginary roots.
D. [tex]\( f(x) \)[/tex] has three real roots.



Answer :

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]