Find the remainder when - (i) `(x^(101)+101)` is divided by `(x+1).` (ii) `(x^(502)+502)` is divided by `(x+1)`.

Explanation

To solve the given problem, we will use the Remainder Theorem. The Remainder Theorem states that if a polynomial \( p(x) \) is divided by \( x - a \), the remainder is \( p(a) \).

### Part (i): Find the remainder when \( x^{101} + 101 \) is divided by \( x + 1 \).

1. **Identify the polynomial and divisor**:
- Let \( p(x) = x^{101} + 101 \).
- We are dividing by \( x + 1 \), which can be rewritten as \( x - (-1) \).

...