11. Always, Sometimes, or Never? a. A, S , or \( \mathrm{N}: \) A 4th degree polynomial can have 2 real roots. b. \( \mathrm{A}, \mathrm{S} \), or \( \mathrm{N}: \) If \( \mathrm{P}(3)=\mathrm{P}(-5)=0 \), then \( \mathrm{P}(\text{x})=\text{x}^{2}+2 \text{x}-15 \).

Let's analyze each statement: ### a. A 4th degree polynomial can have 2 real roots. - **Answer:** **Sometimes (S)** A fourth-degree polynomial can have up to four real roots. It **can** have exactly two real roots (with the other two being complex), but this is not always the case. Therefore, it is **sometimes** possible. ### b. If \( \mathrm{P}(3)=\mathrm{P}(-5)=0 \), then \( \mathrm{P}(\text{x})=\text{x}^{2}+2 \text{x}-15 \). - **Answer:** **Never (N)** If \( \mathrm{P}(3) = 0 \) and \( \mathrm{P}(-5) = 0 \), it means that \( \mathrm{P}(x) \) has factors \( (x - 3) \) and \( (x + 5) \). However, this does **not** necessarily imply that \( \mathrm{P}(x) \) is exactly \( x^{2} + 2x - 15 \). The polynomial could be of higher degree or have additional factors. Therefore, the statement is **never** necessarily true. **Summary of Answers:** - **a. Sometimes (S)** - **b. Never (N)**