If f(x) is a polynomial of degree at least two with integral co-efficients then the remainder when it is divided by(x-a) (x-b) is , where ` a ne b`
(1) ` x[(f(a)-f(b))/(b-a)] + (af(b) -bf(a))/(a-b)` (2) ` x[(f(a) -f(b))/(a-b)] + (af(b) -bf(a))/(a-b)`
(3) `x[(f(b) -f(a))/(a-b)] + (af(b) -bf(a))/(a-b)` (4)`x [ (f(b) -f(a))/(a-b)] + (bf(a) -af(a))/(a-b)`

Answer

To solve the problem, we need to find the remainder when a polynomial \( f(x) \) of degree at least two with integral coefficients is divided by \( (x-a)(x-b) \), where \( a \neq b \).

### Step-by-Step Solution:

1. **Understanding the Remainder**:
When dividing a polynomial \( f(x) \) by a quadratic polynomial \( (x-a)(x-b) \), the remainder \( R(x) \) will be a linear polynomial of the form:
\[ R(x) = cx + d \]
where \( c \) and \( d \) are constants.

2. **Using the Remainder Theorem**:
According to the Remainder Theorem, we can evaluate the polynomial at specific points to find the coefficients \( c \) and \( d \):
- When \( x = a \):
\[ f(a) = R(a) = ca + d \quad \text{(1)} \]
- When \( x = b \):
\[ f(b) = R(b) = cb + d \quad \text{(2)} \]

3. **Setting Up the Equations**:
From equations (1) and (2), we have:
\[ f(a) = ca + d \quad \text{(1)} \]
\[ f(b) = cb + d \quad \text{(2)} \]

4. **Subtracting the Equations**:
Subtract equation (1) from equation (2):
\[ f(b) - f(a) = cb + d - (ca + d) \]
This simplifies to:
\[ f(b) - f(a) = c(b - a) \]
Therefore, we can solve for \( c \):
\[ c = \frac{f(b) - f(a)}{b - a} \]

5. **Finding \( d \)**:
Now, substitute \( c \) back into equation (1) to find \( d \):
\[ f(a) = \left(\frac{f(b) - f(a)}{b - a}\right)a + d \]
Rearranging gives:
\[ d = f(a) - \frac{f(b) - f(a)}{b - a} \cdot a \]
Simplifying this, we get:
\[ d = \frac{af(b) - bf(a)}{b - a} \]

6. **Final Form of the Remainder**:
Thus, the remainder \( R(x) \) can be expressed as:
\[ R(x) = \left(\frac{f(b) - f(a)}{b - a}\right)x + \frac{af(b) - bf(a)}{b - a} \]

### Conclusion:
The remainder when \( f(x) \) is divided by \( (x-a)(x-b) \) is:
\[ R(x) = x\left(\frac{f(b) - f(a)}{b - a}\right) + \frac{af(b) - bf(a)}{b - a} \]