If a polynomial f (x) is divided by ` (x - 3) and (x - 4)` it leaves remainders as 7 and 12 respectively, then find the remainder when f (x) is divided by `(x-3)(x-4)`

Explanation

To solve the problem, we need to find the remainder when the polynomial \( f(x) \) is divided by \( (x - 3)(x - 4) \). Given that the remainders when \( f(x) \) is divided by \( (x - 3) \) and \( (x - 4) \) are 7 and 12 respectively, we can express the remainder in the form \( R(x) = ax + b \).

### Step-by-step Solution:

1. **Understanding the Remainder Theorem**:
According to the Remainder Theorem, if a polynomial \( f(x) \) is divided by \( (x - c) \), the remainder is \( f(c) \).

2. **Setting up the equations**:
We know:
- \( f(3) = 7 \)
- \( f(4) = 12 \)

Since \( f(x) \) can be expressed as:
\[ f(x) = (x - 3)(x - 4)Q(x) + R(x) \]
where \( R(x) = ax + b \) is the remainder when \( f(x) \) is divided by \( (x - 3)(x - 4) \).

3. **Substituting values into the equations**:
We substitute \( x = 3 \) into \( R(x) \):
\[ R(3) = 3a + b = 7 \quad \text{(Equation 1)} \]

Now substitute \( x = 4 \):
\[ R(4) = 4a + b = 12 \quad \text{(Equation 2)} \]

4. **Setting up the system of equations**:
We have the following system of equations:
\[ 3a + b = 7 \quad \text{(1)} \]
\[ 4a + b = 12 \quad \text{(2)} \]

5. **Subtracting the equations**:
Subtract Equation (1) from Equation (2):
\[ (4a + b) - (3a + b) = 12 - 7 \]
This simplifies to:
\[ a = 5 \]

6. **Finding \( b \)**:
Substitute \( a = 5 \) back into Equation (1):
\[ 3(5) + b = 7 \]
\[ 15 + b = 7 \]
\[ b = 7 - 15 = -8 \]

7. **Writing the remainder**:
Now that we have \( a \) and \( b \), we can write the remainder:
\[ R(x) = 5x - 8 \]

### Final Answer:
The remainder when \( f(x) \) is divided by \( (x - 3)(x - 4) \) is:
\[ \boxed{5x - 8} \]