Find, in each case, the remainder when :
(i) `x^(4)- 3x^2 + 2x + 1` is divided by x - 1.
(ii) `x^2 + 3x^2 - 12x + 4` is divided by x - 2.
(iii) `x^4+ 1` is divided by x + 1.

Answer

To find the remainder when the given polynomials are divided by the specified linear factors, we can use the Remainder Theorem. According to this theorem, if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \).

Let's solve each part step by step.

### (i) Find the remainder when \( x^4 - 3x^2 + 2x + 1 \) is divided by \( x - 1 \).

1. **Identify the polynomial and the divisor**:
- Polynomial: \( f(x) = x^4 - 3x^2 + 2x + 1 \)
- Divisor: \( x - 1 \)

2. **Find the value of \( a \)**:
- Set \( x - 1 = 0 \) to find \( a \).
- \( x = 1 \)

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

4. **Conclusion**:
- The remainder when \( x^4 - 3x^2 + 2x + 1 \) is divided by \( x - 1 \) is **1**.

### (ii) Find the remainder when \( x^3 + 3x^2 - 12x + 4 \) is divided by \( x - 2 \).

1. **Identify the polynomial and the divisor**:
- Polynomial: \( f(x) = x^3 + 3x^2 - 12x + 4 \)
- Divisor: \( x - 2 \)

2. **Find the value of \( a \)**:
- Set \( x - 2 = 0 \) to find \( a \).
- \( x = 2 \)

3. **Evaluate \( f(2) \)**:
- Substitute \( x = 2 \) into the polynomial:
\[ f(2) = 2^3 + 3(2^2) - 12(2) + 4 \]
\[ = 8 + 3(4) - 24 + 4 \]
\[ = 8 + 12 - 24 + 4 = 0 \]

4. **Conclusion**:
- The remainder when \( x^3 + 3x^2 - 12x + 4 \) is divided by \( x - 2 \) is **0**.

### (iii) Find the remainder when \( x^4 + 1 \) is divided by \( x + 1 \).

1. **Identify the polynomial and the divisor**:
- Polynomial: \( f(x) = x^4 + 1 \)
- Divisor: \( x + 1 \)

2. **Find the value of \( a \)**:
- Set \( x + 1 = 0 \) to find \( a \).
- \( x = -1 \)

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

4. **Conclusion**:
- The remainder when \( x^4 + 1 \) is divided by \( x + 1 \) is **2**.

### Summary of Remainders:
- (i) Remainder = 1
- (ii) Remainder = 0
- (iii) Remainder = 2