Question
Which of the following shows that polynomials are a closed system under addition?(1 point)
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
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
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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.**
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.**
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