Find the remainder when the polynomial `p(x)=x^(100)-x^(97)+x^(3)` is divided by `x+1` .

Answer

To find the remainder when the polynomial \( p(x) = x^{100} - x^{97} + x^3 \) is divided by \( x + 1 \), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( p(x) \) by \( x - a \) is equal to \( p(a) \).

In our case, since we are dividing by \( x + 1 \), we can rewrite it as \( x - (-1) \). Thus, we need to evaluate \( p(-1) \).

### Step-by-step Solution:

1. **Identify the polynomial and the value to substitute:**
\[ p(x) = x^{100} - x^{97} + x^3 \]
We need to find \( p(-1) \).

2. **Substitute \( x = -1 \) into the polynomial:**
\[ p(-1) = (-1)^{100} - (-1)^{97} + (-1)^3 \]

3. **Calculate each term:**
- \( (-1)^{100} = 1 \) (since 100 is even)
- \( (-1)^{97} = -1 \) (since 97 is odd)
- \( (-1)^3 = -1 \) (since 3 is odd)

4. **Combine the results:**
\[ p(-1) = 1 - (-1) + (-1) \]
Simplifying this:
\[ p(-1) = 1 + 1 - 1 = 1 \]

5. **Conclusion:**
The remainder when \( p(x) \) is divided by \( x + 1 \) is \( 1 \).

### Final Answer:
The remainder is \( 1 \).

---