To determine a factor of the polynomial function [tex]\( f(x) \)[/tex] given its roots, we need to consider the properties of polynomial roots.
Given roots are [tex]\( 3 + \sqrt{5} \)[/tex] and [tex]\(-6 \)[/tex].
First, note that for any polynomial with real coefficients, if [tex]\( 3 + \sqrt{5} \)[/tex] is a root, its conjugate [tex]\( 3 - \sqrt{5} \)[/tex] must also be a root. This is because roots involving square roots must come in conjugate pairs to ensure that the polynomial has real coefficients.
Therefore, the roots of the polynomial function [tex]\( f(x) \)[/tex] we have are:
1. [tex]\( 3 + \sqrt{5} \)[/tex]
2. [tex]\( 3 - \sqrt{5} \)[/tex] (conjugate)
3. [tex]\(-6\)[/tex]
Each of these roots corresponds to a factor of the polynomial. The factor corresponding to the root [tex]\( 3 + \sqrt{5} \)[/tex] is [tex]\((x - (3 + \sqrt{5}))\)[/tex].
Similarly, the factor corresponding to the root [tex]\( 3 - \sqrt{5} \)[/tex] is [tex]\((x - (3 - \sqrt{5}))\)[/tex].
Hence, one of the factors of [tex]\( f(x) \)[/tex] must be [tex]\((x - (3 - \sqrt{5}))\)[/tex].
Given the options:
1. [tex]\((x + (3 - \sqrt{5}))\)[/tex]
2. [tex]\((x - (3 - \sqrt{5}))\)[/tex]
3. [tex]\((x + (5 + \sqrt{3}))\)[/tex]
4. [tex]\((x - (5 - \sqrt{3}))\)[/tex]
The correct answer is:
[tex]\[
(x - (3 - \sqrt{5}))
\][/tex]
