Which of the following shows that polynomials are a closed system under addition?(1 point)

Question

Responses

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=10.994x2+5.96
left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 10.994 x squared plus 5.96

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=3.903x2−12.84x+3.903
left parenthesis 0.923 x squared minus 6.42 x plus 2.98 right parenthesis plus left parenthesis 2.98 minus 6.42 x plus 0.923 x squared right parenthesis equals 3.903 x squared minus 12.84 x plus 3.903

Answer 1

To determine which of the given options correctly shows that polynomials are a closed system under addition, we need to simplify the polynomial expressions shown in both responses.

1. For the first response: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) \]

   Combining like terms:

   • The \(x^2\) terms: \(0.923x^2 + 0.923x^2 = 1.846x^2\)
   • The \(x\) terms: \(-6.42x - 6.42x = -12.84x\)
   • The constant terms: \(2.98 + 2.98 = 5.96\)

   So, it simplifies to: \[ 1.846x^2 - 12.84x + 5.96 \]

2. For the second response: \[ (0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) \]

   Following the same process as above:

   • The \(x^2\) terms: \(0.923x^2 + 0.923x^2 = 1.846x^2\)
   • The \(x\) terms: \(-6.42x - 6.42x = -12.84x\)
   • The constant terms: \(2.98 + 2.98 = 5.96\)

   Therefore, this also simplifies to: \[ 1.846x^2 - 12.84x + 5.96 \]

Since both responses yield the same expression, we'll check the proposed results:

The first response indicates: \[ 10.994 x^2 + 5.96 \]

The second response shows: \[ 3.903 x^2 - 12.84x + 3.903 \]

Since the first response provides an incorrect result of \(10.994 x^2 + 5.96\), that is not valid.

Thus, both options were misleading. If done correctly, the sum of valid polynomials under addition results in a polynomial.

Based on the question regarding closure under addition, we've demonstrated throughout the process that adding two polynomials results in a polynomial.

Since neither is truly viable as a proper demonstration, both should be disregarded thus, neither shows that polynomials are closed under addition effectively. But since we need a response, we could lean on the reasoning behind the first choice figuring if the expression followed a coherent polynomial structure in terms of the closure property.

So the conclusion based on the closure property of polynomials:

Neither option correctly demonstrates closure under polynomial addition due to miscalculation. However, if forced to select, option showing no contradictory values regardless (first option) would be the preferable response if we grounded in polynomial reasoning.

Final answer selection lean: (0.923x^2−6.42x+2.98)+(2.98−6.42x+0.923x^2)=10.994x^2+5.96.