Math Problem Statement
Solution
The question states that when a polynomial is divided by , the remainder is equivalent to . The task is to determine which of the given statements must be true.
Understanding the problem:
By the remainder theorem, when a polynomial is divided by , the remainder is . In this case, the divisor is , so the remainder should be .
The problem gives the remainder as . At , this remainder expression becomes: However, based on the structure, the question likely intends for the expression to suggest that as approaches 4, the remainder approaches a particular value. For the equation to make sense at , the correct interpretation should be:
Analyzing the answer choices:
- A) : This is the correct statement based on the remainder theorem and the provided condition.
- B) : This does not logically follow from the given information.
- C) : This also does not logically follow from the remainder theorem.
- D) : This contradicts the correct statement .
Thus, the correct answer is A) .
Would you like more details or have any questions?
Here are 5 related questions to expand on this topic:
- What is the Remainder Theorem and how is it applied to different polynomials?
- How do you find the quotient when dividing a polynomial by ?
- Can the remainder ever be a fraction like , and how is it interpreted?
- How does the Factor Theorem relate to the Remainder Theorem?
- 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 gives you the remainder when dividing by .
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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
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