If `x^(2)+x+4` is divided by `(x-1)`, find the remainder

Answer

To find the remainder when \( x^2 + x + 4 \) is divided by \( x - 1 \), we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).

### Step-by-Step Solution:

1. **Identify the polynomial and divisor**:
We have the polynomial \( f(x) = x^2 + x + 4 \) and we are dividing by \( x - 1 \).

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, we need to evaluate \( f(1) \) since \( c = 1 \) (from \( x - 1 = 0 \)).

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

4. **Calculate \( f(1) \)**:
\[ f(1) = 1 + 1 + 4 = 6 \]

5. **Conclusion**:
The remainder when \( x^2 + x + 4 \) is divided by \( x - 1 \) is \( 6 \).

### Final Answer:
The remainder is \( 6 \).

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