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The problem asks for the remainder when dividing the polynomial \( f(x) = (x - a)(x - b) \) by another polynomial.

Let's analyze the given options:

### Step 1: Understanding the Problem
Given a polynomial \( f(x) = (x - a)(x - b) \), the problem asks for the remainder when dividing \( f(x) \) by some other polynomial. The choices provided seem to be dealing with expressions that relate to the application of the Remainder Theorem or Polynomial Division.

### Step 2: Apply the Polynomial Remainder Theorem
The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( (x - c) \), the remainder of this division is \( f(c) \). Here, however, the division is more complex, as the options suggest that the polynomial might be divided by linear factors like \( (x-a) \) and \( (x-b) \).

### Step 3: Consider Possible Scenarios for Remainder
1. **Zero Remainder**: If \( f(x) \) is divided by \( (x-a) \) or \( (x-b) \), the remainder would be zero because these are factors of \( f(x) \).
2. **Remainder When Dividing by a Different Polynomial**: The given options hint that a more complex remainder could be involved when dividing by a non-factor polynomial.

### Step 4: Analyze the Provided Options
Without further context, it seems the options might be considering specific cases where \( f(x) \) is divided by polynomials involving both \( (x-a) \) and \( (x-b) \), leading to a possible remainder expression.

### Step 5: Matching with the Right Option
Given that \( f(x) = (x-a)(x-b) \), and considering typical approaches, the remainder in simple polynomial divisions might not be a complex expression.

However, considering the mathematical rigor and likely options, **the correct answer would likely be `E. 0`** because \( (x-a) \) and \( (x-b) \) are the roots of the polynomial. Therefore, the remainder when divided by these factors should be zero.

### Conclusion:
The correct answer is likely **E. 0** based on the given mathematical principles and the nature of the polynomial \( f(x) \).

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**Would you like a detailed explanation or have any other questions?**

Here are 5 related questions:
1. What is the Polynomial Remainder Theorem and how is it used?
2. How do you find the roots of a quadratic polynomial?
3. What are the implications of dividing a polynomial by its factors?
4. How do you perform polynomial long division?
5. What is synthetic division and when is it useful?

**Tip:** Always check if the divisor is a factor of the polynomial; if it is, the remainder should be zero.

Explore how to calculate the remainder when dividing the polynomial (x - a)(x - b) by another polynomial. Understand the application of the Polynomial Remainder Theorem and its implications. Suitable for advanced mathematics students.

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