If the LCM of the polynomials `(x-3)(x-p) and (x+3)(x+5)` is `(x-3)(x+3)(x-p)`, then p is ___________.

Options

A. `-5`
B. `-4`
C. `-2`
D. `-1`

Answer

To find the value of \( p \) in the given problem, we need to analyze the LCM of the polynomials \( (x-3)(x-p) \) and \( (x+3)(x+5) \).

### Step-by-Step Solution:

1. **Understand the Given LCM**:
We are given that the LCM of the polynomials \( (x-3)(x-p) \) and \( (x+3)(x+5) \) is \( (x-3)(x+3)(x-p) \).

2. **Identify the Polynomials**:
The first polynomial is \( (x-3)(x-p) \) and the second polynomial is \( (x+3)(x+5) \).

3. **Factor the LCM**:
The LCM is given as \( (x-3)(x+3)(x-p) \). This means that the LCM contains the factors \( (x-3) \), \( (x+3) \), and \( (x-p) \).

4. **Determine the Factors**:
For the LCM to include \( (x+3) \) and \( (x-p) \), both polynomials must have these factors. Since \( (x-3) \) is a common factor, we check the other polynomial \( (x+3)(x+5) \).

5. **Analyze the Second Polynomial**:
The polynomial \( (x+3)(x+5) \) contains \( (x+3) \) but does not contain \( (x-p) \). For \( (x-p) \) to be included in the LCM, \( p \) must be such that \( (x-p) \) can be derived from the factors of the first polynomial.

6. **Set Up the Condition**:
Since \( (x-p) \) must be a factor of the LCM, we can set \( p \) equal to one of the roots of the second polynomial. The roots of \( (x+3)(x+5) \) are \( -3 \) and \( -5 \).

7. **Choose the Root**:
Since the LCM includes \( (x-p) \) and \( (x+3) \) is already included in the LCM, we can set \( p = -5 \) to satisfy the condition of the LCM.

8. **Conclusion**:
Therefore, the value of \( p \) is \( -5 \).

### Final Answer:
The value of \( p \) is \( -5 \).