Can `(x - 1)` be the remainder on division of a polynomial `P(x)` by `2x + 3`? Justify your answer.

Answer

To determine whether \( (x - 1) \) can be the remainder when dividing a polynomial \( P(x) \) by \( 2x + 3 \), we need to consider the properties of polynomial division, particularly the degree of the remainder.

### Step-by-Step Solution:

1. **Identify the Degree of the Divisor**:
The polynomial we are dividing by is \( 2x + 3 \). The degree of this polynomial is 1 (since the highest power of \( x \) is 1).

2. **Understand the Remainder Theorem**:
According to the Remainder Theorem, when a polynomial \( P(x) \) is divided by a linear polynomial \( ax + b \), the remainder will be a polynomial of degree less than the degree of the divisor. Since the divisor \( 2x + 3 \) has a degree of 1, the remainder must have a degree less than 1.

3. **Determine the Degree of the Proposed Remainder**:
The proposed remainder is \( (x - 1) \). The degree of \( (x - 1) \) is also 1 (the highest power of \( x \) is 1).

4. **Compare the Degrees**:
Since the degree of the remainder \( (x - 1) \) is equal to the degree of the divisor \( (2x + 3) \), it violates the condition that the degree of the remainder must be less than the degree of the divisor.

5. **Conclusion**:
Therefore, \( (x - 1) \) cannot be the remainder when dividing a polynomial \( P(x) \) by \( 2x + 3 \).

### Final Answer:
No, \( (x - 1) \) cannot be the remainder on division of a polynomial \( P(x) \) by \( 2x + 3 \) because the degree of the remainder must be less than the degree of the divisor.