To determine which of the provided options shows that polynomials are a closed system under addition, we need to verify if the sum of two polynomials is also a polynomial.
The form of the addition you provided is:
\[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) \]
Let's analyze each response one by one.
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First Response: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 3.903x^2 - 12.84x + 3.903 \]
- This simplifies to a polynomial, confirming closure.
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Second Response: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 10.994x^2 + 5.96 \]
- This also simplifies correctly to a polynomial.
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Third Response: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 2x^2 - 13x + 6 \]
- This is also a valid polynomial.
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Fourth Response: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96 \]
- This too simplifies correctly to a polynomial.
Since all of the responses show valid polynomial results from the addition of two polynomials, they all, in fact, demonstrate that polynomials are closed under addition.
However, to pick the most correct example that accurately displays arithmetic with polynomials, we can compute the first response, which correctly combines the polynomials:
\[ (0.923x^2 + 0.923x^2) + (-6.42x - 6.42x) + (2.98 + 2.98) = 1.846x^2 - 12.84x + 5.96 \]
Each response ends up being correct in its own computation; however, based on standard polynomial addition, the first one is a straightforward and simple computation.
Make sure to verify all calculations if you are uncertain about a specific possibility! Selecting the first option maintains clarity and correctness regarding polynomial addition.