To determine the required additional root for the polynomial function [tex]\( f(x) \)[/tex] with rational coefficients, let's start by understanding the properties of polynomials with rational coefficients.
1. Roots and Rational Coefficients:
- If a polynomial has rational coefficients and has an irrational root, the conjugate of that irrational root must also be a root of the polynomial.
- This means, for any root of the form [tex]\( a + \sqrt{b} \)[/tex], the root [tex]\( a - \sqrt{b} \)[/tex] must also be present if the polynomial is to have rational coefficients.
2. Given Roots:
- The polynomial [tex]\( f(x) \)[/tex] has the roots: [tex]\( 0 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 3 + \sqrt{11} \)[/tex].
3. Finding the Conjugate Root:
- Since [tex]\( 3 + \sqrt{11} \)[/tex] is an irrational root, its conjugate [tex]\( 3 - \sqrt{11} \)[/tex] must also be a root to ensure the polynomial has rational coefficients.
4. Answer:
- Therefore, the root that must also be a root of [tex]\( f(x) \)[/tex] is [tex]\( 3 - \sqrt{11} \)[/tex].
Hence, the correct answer is [tex]\( 3 - \sqrt{11} \)[/tex].
