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The question states that when a polynomial \( f(x) \) is divided by \( x - 4 \), the remainder is equivalent to \( \frac{5}{x-4} \). The task is to determine which of the given statements must be true.

### Understanding the problem:
By the **remainder theorem**, when a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). In this case, the divisor is \( x - 4 \), so the remainder should be \( f(4) \).

The problem gives the remainder as \( \frac{5}{x - 4} \). At \( x = 4 \), this remainder expression becomes:
\[
\frac{5}{x - 4} \quad \text{becomes undefined at} \quad x = 4.
\]
However, based on the structure, the question likely intends for the expression to suggest that as \( x \) approaches 4, the remainder approaches a particular value. For the equation to make sense at \( x = 4 \), the correct interpretation should be:

\[
f(4) = 5.
\]

### Analyzing the answer choices:
- **A) \( f(4) = 5 \)**: This is the correct statement based on the remainder theorem and the provided condition.
- **B) \( f(-4) = 5 \)**: This does not logically follow from the given information.
- **C) \( f(5) = 4 \)**: This also does not logically follow from the remainder theorem.
- **D) \( f(4) = 0 \)**: This contradicts the correct statement \( f(4) = 5 \).

Thus, **the correct answer is A) \( f(4) = 5 \)**. 

Would you like more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. What is the Remainder Theorem and how is it applied to different polynomials?
2. How do you find the quotient when dividing a polynomial by \( x - a \)?
3. Can the remainder ever be a fraction like \( \frac{5}{x-4} \), and how is it interpreted?
4. How does the Factor Theorem relate to the Remainder Theorem?
5. What happens if the divisor is a quadratic expression instead of a linear one?

**Tip:** The Remainder Theorem is a powerful tool when evaluating polynomials at specific points. Always remember that \( f(a) \) gives you the remainder when dividing by \( x - a \).

When the polynomial f(x) is divided by x - 4, the remainder is equivalent to 5 / (x - 4). Which of the following statements must be true?

Explore how to solve a polynomial division problem using the Remainder Theorem. The task involves determining which statement is true when a polynomial f(x) is divided by x - 4, and the remainder is 5 / (x - 4). Suitable for Grades 9-12.

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