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)=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

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=1.846x2−12.84x+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 1.846 x squared minus 12.84 x plus 5.96

(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)=2x2−13x+6

Answer 1

To determine which option shows that polynomials are a closed system under addition, we need to look for an expression where the sum of two polynomials is itself a polynomial.

The main polynomial expressions involved are:
1. \(0.923x^2 - 6.42x + 2.98\)
2. \(2.98 - 6.42x + 0.923x^2\)

Now, let's add them up:

1. The coefficient of \(x^2\):
\[
0.923 + 0.923 = 1.846
\]

2. The coefficient of \(x\):
\[
-6.42 - 6.42 = -12.84
\]

3. The constant term:
\[
2.98 + 2.98 = 5.96
\]

Thus, the result of adding those two polynomials is:
\[
1.846x^2 - 12.84x + 5.96
\]

Now we compare this result with the options presented:

- **1**: \(3.903x^2 - 12.84x + 3.903\) (incorrect)
- **2**: \(1.846x^2 - 12.84x + 5.96\) (correct)
- **3**: \(10.994x^2 + 5.96\) (incorrect)
- **4**: \(2x^2 - 13x + 6\) (incorrect)

Therefore, the correct response that shows that polynomials are closed under addition is:

**(0.923x^2−6.42x+2.98)+(2.98−6.42x+0.923x^2)=1.846x^2−12.84x+5.96**.