What does it mean that polynomials are closed under addition?
Understand the Problem
The question is asking about the concept of closure under addition for polynomials. Specifically, it seeks to identify which statement correctly describes what it means for polynomials to be closed under addition.
Answer
The result of adding two polynomials is always a polynomial.
Answer for screen readers
The correct answer is: The result of adding two polynomials is always a polynomial.
Steps to Solve
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Understand Closure Under Addition Closure under addition means that when you add two elements of a set, the result is also an element of that set. For polynomials, this implies that if you add two polynomials, the result must also be a polynomial.
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Evaluate Each Option Look at each potential answer to see if it correctly describes closure under addition for polynomials.
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Analyze the First Option The statement "The result of adding two polynomials is always a polynomial" aligns with the definition of closure under addition, as it states that the sum of any two polynomials is still a polynomial.
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Analyze the Second Option "Adding polynomials never results in another polynomial" contradicts the definition and is incorrect.
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Analyze the Third Option "Polynomials can be added, but they cannot be subtracted" is misleading. Polynomials can both be added and subtracted, so this option is incorrect.
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Analyze the Fourth Option "If you add polynomials, the result may or may not be a polynomial" contradicts the definition of closure and is therefore incorrect.
The correct answer is: The result of adding two polynomials is always a polynomial.
More Information
Closure under addition for polynomials is a fundamental concept in algebra, emphasizing that the set of polynomials is closed under addition, meaning that combining polynomials will always yield another polynomial.
Tips
- Confusing closure under addition with closure under other operations like subtraction or multiplication. It is essential to focus only on the specific operation being discussed.
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