Math Problem Statement

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?

Solution

The question states that when a polynomial f(x)f(x) is divided by x4x - 4, the remainder is equivalent to 5x4\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)f(x) is divided by xax - a, the remainder is f(a)f(a). In this case, the divisor is x4x - 4, so the remainder should be f(4)f(4).

The problem gives the remainder as 5x4\frac{5}{x - 4}. At x=4x = 4, this remainder expression becomes: 5x4becomes undefined atx=4.\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 xx approaches 4, the remainder approaches a particular value. For the equation to make sense at x=4x = 4, the correct interpretation should be:

f(4)=5.f(4) = 5.

Analyzing the answer choices:

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

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

Would you like more details or have any questions?


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 xax - a?
  3. Can the remainder ever be a fraction like 5x4\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)f(a) gives you the remainder when dividing by xax - a.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Remainder Theorem

Formulas

Remainder Theorem: f(a) gives the remainder when dividing f(x) by x - a

Theorems

Remainder Theorem

Suitable Grade Level

Grades 9-12