If `x^(11) +3` is divided by `(x+1)`, find the remainder

Answer

To find the remainder when \( x^{11} + 3 \) is divided by \( x + 1 \), we can use the Remainder Theorem. The Remainder Theorem states that the remainder of a polynomial \( f(x) \) when divided by \( x - c \) is equal to \( f(c) \). In this case, we need to find \( f(-1) \) since we are dividing by \( x + 1 \) (which can be rewritten as \( x - (-1) \)).

### Step-by-Step Solution:

1. **Define the polynomial function**:
Let \( f(x) = x^{11} + 3 \).

2. **Evaluate the function at \( x = -1 \)**:
We need to find \( f(-1) \):
\[ f(-1) = (-1)^{11} + 3 \]

3. **Calculate \( (-1)^{11} \)**:
Since 11 is an odd number, \( (-1)^{11} = -1 \).

4. **Substitute back into the function**:
Now substitute this value back into the equation:
\[ f(-1) = -1 + 3 \]

5. **Simplify the result**:
\[ f(-1) = 2 \]

6. **State the remainder**:
Therefore, the remainder when \( x^{11} + 3 \) is divided by \( x + 1 \) is \( 2 \).

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

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