Patricia is studying a polynomial function [tex]f(x)[/tex]. Three given roots of [tex]f(x)[/tex] are [tex]-11-\sqrt{2}, 3+4i[/tex], and 10. Patricia concludes that [tex]f(x)[/tex] must be a polynomial with degree 4. Which statement is true?

A. Patricia is correct because [tex]-11+\sqrt{2}[/tex] must be a root.
B. Patricia is correct because [tex]3-4i[/tex] must be a root.
C. Patricia is not correct because both [tex]3-4i[/tex] and [tex]11+\sqrt{2}[/tex] must be roots.
D. Patricia is not correct because both [tex]3-4i[/tex] and [tex]-11+\sqrt{2}[/tex] must be roots.



Answer :

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.

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