When `(x^4-3x^3+2x^2-5x+7)` is divided by (x-1) then the remainder is

Options

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

Answer

To find the remainder when the polynomial \( f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7 \) is divided by \( x - 1 \), we can use the Remainder Theorem. According to the 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**:
- The polynomial is \( f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7 \).
- The divisor is \( x - 1 \).

2. **Apply the Remainder Theorem**:
- According to the Remainder Theorem, we need to evaluate \( f(1) \) since we are dividing by \( x - 1 \).

3. **Calculate \( f(1) \)**:
- Substitute \( x = 1 \) into the polynomial:
\[ f(1) = (1)^4 - 3(1)^3 + 2(1)^2 - 5(1) + 7 \]

4. **Simplify the expression**:
- Calculate each term:
\[ = 1 - 3 + 2 - 5 + 7 \]
- Combine the terms step by step:
\[ = (1 - 3) + (2 - 5) + 7 \]
\[ = -2 - 3 + 7 \]
\[ = -5 + 7 \]
\[ = 2 \]

5. **Conclusion**:
- The remainder when \( f(x) \) is divided by \( x - 1 \) is \( 2 \).

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