To determine the best completion for the sentence "If you know a root of a function is [tex]\( -2 + \sqrt{3} i \)[/tex], then [tex]\(\qquad\)[/tex]", follow these steps:
1. Understand the Concept of Roots and Complex Conjugates:
- For polynomials with real coefficients, if you have a complex root [tex]\( a + bi \)[/tex], then its complex conjugate [tex]\( a - bi \)[/tex] must also be a root. This is because complex roots for polynomials with real coefficients always come in conjugate pairs.
2. Application of the Concept:
- Given that [tex]\( -2 + \sqrt{3} i \)[/tex] is a root and considering our polynomial has real coefficients, its complex conjugate [tex]\( -2 - \sqrt{3} i \)[/tex] must also be a root of the function.
3. Determine if the Conjugate is a Known or Possible Root:
- Since the mathematical property guarantees [tex]\( -2 - \sqrt{3} i \)[/tex] as a root due to the nature of polynomials with real coefficients, it is not just a possible root but a known root.
4. Revisiting the Provided Options:
- [tex]\( 2 + \sqrt{3} i \)[/tex] is neither a possible nor known root as it does not relate directly via conjugation with [tex]\( -2 + \sqrt{3} i \)[/tex].
- [tex]\( -2 - \sqrt{3} i \)[/tex] is directly the complex conjugate of the given root [tex]\( -2 + \sqrt{3} i \)[/tex], thus, it is a known root due to the properties discussed.
Therefore, the correct completion for the sentence is:
[tex]\[ -2 - \sqrt{3} i \text{ is a known root.} \][/tex]
This conclusion aligns directly with the confirmed results.
