Patricia is studying a polynomial function f(x). Three given roots of f(x) are Negative 11 minus StartRoot 2 EndRoot i, 3 + 4i, and 10. Patricia concludes that f(x) must be a polynomial with degree 4. Which statement is true?
Patricia is correct because Negative 11 + StartRoot 2 EndRoot i must be a root.
Patricia is correct because 3 – 4i must be a root.
Patricia is not correct because both 3 – 4i  and 11 + StartRoot 2 EndRoot i must be roots.
Patricia is not correct because both 3 – 4i and Negative 11 + StartRoot 2 EndRoot i must be roots.

To determine whether Patricia's conclusion is correct, let's analyze the situation:

### Given:
- Roots of \( f(x) \): 
  1. \( -11 - \sqrt{2}i \)
  2. \( 3 + 4i \)
  3. \( 10 \)

### Key Concept:
- **Complex roots** of a polynomial with real coefficients always occur in **conjugate pairs**. This means that if \( a + bi \) is a root, then \( a - bi \) must also be a root.

### Step-by-Step Analysis:

1. Since one of the roots is \( -11 - \sqrt{2}i \), its conjugate \( -11 + \sqrt{2}i \) must also be a root (for the polynomial to have real coefficients).

2. Similarly, since \( 3 + 4i \) is a root, its conjugate \( 3 - 4i \) must also be a root.

3. The root \( 10 \) is real and doesn't require a conjugate pair.

### Conclusion:
- The polynomial must have the following roots: \( -11 - \sqrt{2}i \), \( -11 + \sqrt{2}i \), \( 3 + 4i \), and \( 3 - 4i \).
- Therefore, Patricia's polynomial has **5** roots: \( -11 - \sqrt{2}i \), \( -11 + \sqrt{2}i \), \( 3 + 4i \), \( 3 - 4i \), and \( 10 \), but a polynomial with real coefficients must only have **4 distinct roots**, not 5.

Thus, the correct statement is:
- **Patricia is not correct because both 3 – 4i and Negative 11 + StartRoot 2 EndRoot i must be roots.**

### Questions for further exploration:
1. What happens if a polynomial has an odd degree?
2. How can you find the conjugates of complex roots?
3. What is the minimum degree of a polynomial with only real coefficients that has two distinct complex roots?
4. How do you construct a polynomial from its roots?
5. How do polynomials with irrational or complex coefficients differ from those with real coefficients?

**Tip**: Always remember, when dealing with polynomials with real coefficients, complex roots come in conjugate pairs!

Patricia is studying a polynomial function f(x). Three given roots of f(x) are -11 - √2i, 3 + 4i, and 10. Patricia concludes that f(x) must be a polynomial with degree 4. Which statement is true?
Patricia is correct because -11 + √2i must be a root.
Patricia is correct because 3 – 4i must be a root.
Patricia is not correct because both 3 – 4i and 11 + √2i must be roots.
Patricia is not correct because both 3 – 4i and -11 + √2i must be roots.

Explore Patricia's polynomial problem, involving complex roots and conjugate pairs. Understand why polynomials with real coefficients must have complex roots occur in pairs and learn how to determine the correct degree of the polynomial. Suitable for students in Grades 10-12.

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