To determine if Patricia's conclusion is correct, let's analyze the polynomial function [tex]\( f(x) \)[/tex] and its roots in detail.
1. Given Roots:
- [tex]\( -11 - \sqrt{2} \)[/tex]
- [tex]\( 3 + 4i \)[/tex]
- [tex]\( 10 \)[/tex]
2. Roots and Complex Conjugates Property:
- If a polynomial has real coefficients, any complex roots must occur in conjugate pairs. This means if [tex]\( z \)[/tex] is a root, then its complex conjugate [tex]\( \bar{z} \)[/tex] must also be a root.
- For polynomial functions with real coefficients, irrational roots involving a square root must also occur in conjugate pairs for the same reason.
3. Identifying Conjugate Roots:
- Since [tex]\( f(x) \)[/tex] has a root [tex]\( 3 + 4i \)[/tex], its complex conjugate [tex]\( 3 - 4i \)[/tex] must also be a root because [tex]\( f(x) \)[/tex] needs real coefficients.
- Similarly, since [tex]\( f(x) \)[/tex] has a root [tex]\( -11 - \sqrt{2} \)[/tex], its conjugate [tex]\( -11 + \sqrt{2} \)[/tex] must also be a root in order to maintain real coefficients.
4. Determining Degree of the Polynomial:
- Given roots: [tex]\( 10 \)[/tex] (1 real), [tex]\( -11 - \sqrt{2} \)[/tex] and [tex]\( -11 + \sqrt{2} \)[/tex] (irrational conjugates), [tex]\( 3 + 4i \)[/tex] and [tex]\( 3 - 4i \)[/tex] (complex conjugates).
- These all together provide a total of 4 roots.
Thus, the polynomial [tex]\( f(x) \)[/tex] must be at least degree 4 to accommodate all these roots.
5. True Statement:
- Based on the roots and their necessary conjugates, Patricia is correct because the conjugate pairs [tex]\( 3 - 4i \)[/tex] and [tex]\( -11 + \sqrt{2} \)[/tex] must also be roots.
Therefore, the correct statement is:
Patricia is correct because both [tex]\( 3-4 i \)[/tex] and [tex]\( -11+\sqrt{2} \)[/tex] must be roots.
