When working with polynomial functions that have rational coefficients, an important property to consider is that any non-rational root must appear in conjugate pairs. This is necessary to ensure that the polynomial retains rational coefficients.
Given the roots of the polynomial function [tex]\( f(x) \)[/tex]:
1. [tex]\(0\)[/tex]
2. [tex]\(4\)[/tex]
3. [tex]\(3 + \sqrt{11}\)[/tex]
Let’s analyze the roots step by step:
- [tex]\(0\)[/tex] and [tex]\(4\)[/tex] are rational numbers, so they do not affect the rationality of the polynomial's coefficients.
- However, [tex]\(3 + \sqrt{11}\)[/tex] is not a rational number. Since the polynomial has rational coefficients, the root [tex]\( 3 + \sqrt{11} \)[/tex] must be paired with its conjugate to ensure that all coefficients remain rational.
The conjugate of [tex]\( 3 + \sqrt{11} \)[/tex] is [tex]\( 3 - \sqrt{11} \)[/tex].
Therefore, by the principle that non-rational roots must appear in conjugate pairs, the polynomial function [tex]\( f(x) \)[/tex] must also have the root [tex]\( 3 - \sqrt{11} \)[/tex].
Thus, the correct answer is:
[tex]\[ 3 - \sqrt{11} \][/tex]
