To determine which of the given options correctly shows that polynomials are a closed system under addition, we need to perform the addition on the left-hand side for each option and compare it to the result on the right-hand side.
Let's evaluate the addition of the two polynomials in each option:
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Option 1: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) \] Combine like terms:
- For \(x^2\): \(0.923x^2 + 0.923x^2 = 1.846x^2\)
- For \(x\): \(-6.42x - 6.42x = -12.84x\)
- Constant: \(2.98 + 2.98 = 5.96\) Resulting polynomial: \[ 1.846x^2 - 12.84x + 5.96 \] So this does not match \(3.903x^2 - 12.84x + 3.903\).
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Option 2: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) \] We already did this and found the result to be \[ 1.846x^2 - 12.84x + 5.96 \] So this does not match \(1.846x^2 - 12.84x + 5.96\).
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Option 3: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) \] We already evaluated this, and it equals \[ 1.846x^2 - 12.84x + 5.96 \] So this does not match \(2x^2 - 13x + 6\).
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Option 4: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) \] As established, the result is \[ 1.846x^2 - 12.84x + 5.96 \] So this does not match \(10.994x^2 + 5.96\).
After evaluating all options, none of the provided options show the addition correctly that would confirm polynomials are closed under addition, because the computations lead to \(1.846x^2 - 12.84x + 5.96\) consistently across all cases, but none of the right-hand results corresponded to that result correctly. As such, none of the given options are correct reflections of polynomial addition leading to a valid polynomial expression.