The remainder when `a^(3) + 2a^(2) - 3a + 2` is divided by a - 1 is -

Options

A. 2
B. `-2`
C. 1
D. 0

Answer

To find the remainder when \( a^3 + 2a^2 - 3a + 2 \) is divided by \( a - 1 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of a polynomial \( f(a) \) when divided by \( a - c \) is equal to \( f(c) \).

### Step-by-step Solution:

1. **Identify the polynomial and the divisor**:
We have the polynomial \( f(a) = a^3 + 2a^2 - 3a + 2 \) and we are dividing by \( a - 1 \).

2. **Apply the Remainder Theorem**:
We need to evaluate \( f(1) \) because we are dividing by \( a - 1 \) (where \( c = 1 \)).

3. **Substitute \( a = 1 \) into the polynomial**:
\[ f(1) = (1)^3 + 2(1)^2 - 3(1) + 2 \]

4. **Calculate each term**:
- \( (1)^3 = 1 \)
- \( 2(1)^2 = 2 \)
- \( -3(1) = -3 \)
- The constant term is \( +2 \)

5. **Combine the results**:
\[ f(1) = 1 + 2 - 3 + 2 \]
\[ = 1 + 2 = 3 \]
\[ = 3 - 3 = 0 \]
\[ = 0 + 2 = 2 \]

6. **Conclusion**:
The remainder when \( a^3 + 2a^2 - 3a + 2 \) is divided by \( a - 1 \) is \( 2 \).

### Final Answer:
The remainder is \( 2 \).