To determine which response correctly shows that polynomials are a closed system under addition, we should verify the correctness of the addition in each response. The concept of closure under addition means that when you add two polynomials, the result should also be a polynomial.
Let's break down the first polynomial:
- \(0.923x^2 - 6.42x + 2.98\)
- \(2.98 - 6.42x + 0.923x^2\)
Now let's add these two polynomials together:
\[ (0.923x^2 - 6.42x + 2.98) + (0.923x^2 - 6.42x + 2.98) \]
Combine like terms:
- For \(x^2\) terms: \(0.923x^2 + 0.923x^2 = 1.846x^2\)
- For \(x\) terms: \(-6.42x - 6.42x = -12.84x\)
- For constant terms: \(2.98 + 2.98 = 5.96\)
Thus, the result is:
\[ 1.846x^2 - 12.84x + 5.96 \]
The first response is correct:
\[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96 \]
Now, let's briefly check the other responses:
- Second response: \(10.994x^2 + 5.96\): This does not correctly combine the \(x^2\) terms.
- Third response: \(3.903x^2 - 12.84x + 3.903\): This also fails to combine \(x^2\) terms correctly.
- Fourth response: \(2x^2 - 13x + 6\): This is incorrect as well.
Therefore, the only response that shows that polynomials are closed under addition is the first one:
Correct Response: (0.923x^2−6.42x+2.98)+(2.98−6.42x+0.923x^2)=1.846x^2−12.84x+5.96