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The problem in the image is asking to find the remainder when a function \( f(x) \) is divided by \( (x-3)^2(x-4) \), given certain information about the remainders when \( f(x) \) is divided by \( (x-3)^2 \) and \( x-4 \).

Here’s the key information extracted from the image:
- When \( f(x) \) is divided by \( (x-3)^2 \), the remainder is \( 8x - 24 \).
- When \( f(x) \) is divided by \( x - 4 \), the remainder is 20.

We are asked to find the remainder when \( f(x) \) is divided by \( (x-3)^2(x-4) \).

### Step-by-Step Approach:
1. **Remainder Theorem Setup**:
   By the polynomial remainder theorem, when \( f(x) \) is divided by \( (x - 3)^2(x - 4) \), the remainder will be of the form:
   \[
   R(x) = A(x - 4) + B
   \]
   where \( A \) and \( B \) are constants to be determined based on the given information.

2. **Use Information from Divisions**:
   - From the first condition, when \( f(x) \) is divided by \( (x - 3)^2 \), the remainder is \( 8x - 24 \). This implies:
     \[
     R(x) = 8x - 24 \quad \text{for} \quad (x - 3)^2
     \]
   - From the second condition, when \( f(x) \) is divided by \( x - 4 \), the remainder is 20. This implies:
     \[
     R(4) = 20
     \]

3. **Solving for Remainder**:
   Combine both pieces of information to determine the full remainder when dividing \( f(x) \) by \( (x - 3)^2(x - 4) \).

Would you like more detailed explanations or any further clarification on the steps so far?

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### Follow-up Questions:
1. What is the remainder when dividing \( f(x) \) by \( (x-4)^2 \)?
2. How can the Remainder Theorem help in similar polynomial division problems?
3. How does the method change if we divide \( f(x) \) by a product of three linear factors?
4. Can you explain how the degrees of divisors affect the structure of the remainder?
5. What are some practical applications of polynomial division in calculus?

**Tip**: Always analyze the degree of the divisor when using the Remainder Theorem, as it helps in structuring the remainder accurately.

Find the remainder when f(x) is divided by (x-3)^2(x-4), given that when f(x) is divided by (x-3)^2, the remainder is 8x - 24, and when f(x) is divided by x - 4, the remainder is 20.

Explore how to find the remainder when a function f(x) is divided by (x-3)^2(x-4) using the Remainder Theorem. Learn the step-by-step solution, applying algebraic concepts and polynomial division techniques suitable for Grades 11-12.

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