The polynomials `(x^(3)-1)` and `(x^(2)+1)` are divided by `(x+1)` leave the reminder as `R_(1)` and `R_(2)` respectively. The true statement among the following is

Options

A. `R_(1)+R_(2)=0`
B. `R_(1)-R_(2)=0`
C. `2R_(1)+R_(2)=0`
D. `R_(1)-2R_(2)=0`

Answer

To solve the problem, we need to find the remainders \( R_1 \) and \( R_2 \) when the polynomials \( P_1(x) = x^3 - 1 \) and \( P_2(x) = x^2 + 1 \) are divided by \( x + 1 \).

### Step-by-Step Solution:

1. **Identify the Polynomials**:
- Let \( P_1(x) = x^3 - 1 \)
- Let \( P_2(x) = x^2 + 1 \)

2. **Evaluate \( R_1 \)**:
- To find \( R_1 \), we substitute \( x = -1 \) into \( P_1(x) \):
\[ R_1 = P_1(-1) = (-1)^3 - 1 = -1 - 1 = -2 \]

3. **Evaluate \( R_2 \)**:
- To find \( R_2 \), we substitute \( x = -1 \) into \( P_2(x) \):
\[ R_2 = P_2(-1) = (-1)^2 + 1 = 1 + 1 = 2 \]

4. **Check the Relationship**:
- Now we check the sum \( R_1 + R_2 \):
\[ R_1 + R_2 = -2 + 2 = 0 \]

5. **Conclusion**:
- The true statement among the options is that \( R_1 + R_2 = 0 \).