When `(x^(4)-3x^(3)+2x^(2)-5x+7)` is divided by `(x-2)`, then the remainder is :

Options

A. 0
B. `-3`
C. 2
D. 3

Answer

To find the remainder when \( f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7 \) is divided by \( x - 2 \), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
We have \( f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7 \) and we are dividing by \( x - 2 \).

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, we need to evaluate \( f(2) \) to find the remainder.

3. **Substitute \( x = 2 \) into the polynomial**:
\[ f(2) = (2^4) - 3(2^3) + 2(2^2) - 5(2) + 7 \]

4. **Calculate each term**:
- \( 2^4 = 16 \)
- \( 3(2^3) = 3 \times 8 = 24 \)
- \( 2(2^2) = 2 \times 4 = 8 \)
- \( 5(2) = 10 \)

5. **Combine the results**:
\[ f(2) = 16 - 24 + 8 - 10 + 7 \]

6. **Perform the arithmetic**:
- First, calculate \( 16 - 24 = -8 \)
- Then, \( -8 + 8 = 0 \)
- Next, \( 0 - 10 = -10 \)
- Finally, \( -10 + 7 = -3 \)

7. **Conclusion**:
Therefore, the remainder when \( f(x) \) is divided by \( x - 2 \) is \( -3 \).

### Final Answer:
The remainder is \( -3 \).