If `x^(2) + bx + 7` is divided by `(x-1)` leaves remainder 12 find b?

Answer

To solve the problem, we need to find the value of \( b \) such that when the polynomial \( f(x) = x^2 + bx + 7 \) is divided by \( x - 1 \), it leaves a remainder of 12.

### Step-by-step Solution:

1. **Identify the polynomial and the divisor**:
We have the polynomial \( f(x) = x^2 + bx + 7 \) and we are dividing it by \( x - 1 \).

2. **Use the Remainder Theorem**:
According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). In this case, \( c = 1 \).

3. **Calculate \( f(1) \)**:
We need to find \( f(1) \):
\[ f(1) = 1^2 + b \cdot 1 + 7 \]
Simplifying this gives:
\[ f(1) = 1 + b + 7 = b + 8 \]

4. **Set the equation equal to the remainder**:
We know from the problem that the remainder is 12. Therefore, we set up the equation:
\[ b + 8 = 12 \]

5. **Solve for \( b \)**:
To find \( b \), we can rearrange the equation:
\[ b = 12 - 8 \]
Simplifying this gives:
\[ b = 4 \]

### Final Answer:
The value of \( b \) is \( 4 \).

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