To solve this problem, we need to understand the property of polynomials with rational coefficients and how their roots behave, especially when they include irrational numbers like [tex]\( \sqrt{11} \)[/tex].
Step 1: Identify the given roots - We know that the given roots of the polynomial are:
- [tex]\( 0 \)[/tex]
- [tex]\( 4 \)[/tex]
- [tex]\( 3 + \sqrt{11} \)[/tex]
Step 2: Recall the principle - For polynomials with rational coefficients, if a non-rational root [tex]\( a + b\sqrt{c} \)[/tex] exists, its conjugate [tex]\( a - b\sqrt{c} \)[/tex] must also be a root to ensure the polynomial coefficients remain rational. This is because any irrational parts need to cancel out when forming the polynomial equation.
Step 3: Identify the conjugate root - The conjugate of the root [tex]\( 3 + \sqrt{11} \)[/tex] is [tex]\( 3 - \sqrt{11} \)[/tex].
Step 4: Conclusion - Therefore, the root [tex]\( 3 - \sqrt{11} \)[/tex] must also be a root of the polynomial function [tex]\( f(x) \)[/tex].
So, the correct answer is:
[tex]\[ 3 - \sqrt{11} \][/tex]
This matches our choice:
[tex]\[ \boxed{3 - \sqrt{11}} \][/tex]
