For a polynomial `p (x)`, the value of p(3) is `−2`.
Which of the following must be true about `p (x)` ?

Options

A. `x − 5` is a factor of `p (x)`.
B. `x − 2` is a factor of `p(x)`.
C. `x + 2` is a factor of `p (x)`.
D. The remainder when `p (x)` is divided by `x − 3` is` −2.`

Answer

To solve the problem, we need to analyze the information given about the polynomial \( p(x) \) and the value \( p(3) = -2 \). We will evaluate the options provided to determine which must be true about \( p(x) \).

### Step-by-Step Solution:

1. **Understanding the Polynomial and Remainder Theorem**:
The Remainder Theorem states that if a polynomial \( p(x) \) is divided by \( x - a \), the remainder of this division is \( p(a) \). In this case, since we know \( p(3) = -2 \), we can apply this theorem.

2. **Applying the Remainder Theorem**:
Since \( p(3) = -2 \), we can conclude that when \( p(x) \) is divided by \( x - 3 \), the remainder is \( -2 \). This means:
\[ p(x) = (x - 3)q(x) - 2 \]
for some polynomial \( q(x) \).

3. **Evaluating the Options**:
- **Option 1**: \( x - 5 \) is a factor of \( p(x) \).
- This is not necessarily true since we have no information about \( p(5) \).
- **Option 2**: \( x - 2 \) is a factor of \( p(x) \).
- Again, we have no information about \( p(2) \), so this cannot be concluded.
- **Option 3**: \( x + 2 \) is a factor of \( p(x) \).
- There is no information about \( p(-2) \), so this cannot be concluded either.
- **Option 4**: The remainder when \( p(x) \) is divided by \( x - 3 \) is \( -2 \).
- This is true based on our application of the Remainder Theorem.

4. **Conclusion**:
The only statement that must be true about \( p(x) \) given \( p(3) = -2 \) is that the remainder when \( p(x) \) is divided by \( x - 3 \) is \( -2 \). Therefore, the correct answer is **Option 4**.