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
open paren 0 point 9 2 3 x squared minus 6 point 4 2 x plus 2 point 9 8 close paren plus open paren 2 point 9 8 minus 6 point 4 2 x plus 0 point 9 2 3 x squared close paren is equal to 3 point 9 0 3 x squared minus 12 point 8 4 x plus 3 point 9 0 3 - no response given

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=10.994x2+5.96
open paren 0 point 9 2 3 x squared minus 6 point 4 2 x plus 2 point 9 8 close paren plus open paren 2 point 9 8 minus 6 point 4 2 x plus 0 point 9 2 3 x squared close paren is equal to 10 point 9 9 4 x squared plus 5 point 9 6 - incorrect

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=1.846x2−12.84x+5.96
open paren 0 point 9 2 3 x squared minus 6 point 4 2 x plus 2 point 9 8 close paren plus open paren 2 point 9 8 minus 6 point 4 2 x plus 0 point 9 2 3 x squared close paren is equal to 1 point 8 4 6 x squared minus 12 point 8 4 x plus 5 point 9 6 - no response given

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=2x2−13x+6

Answer 1

To determine which of the given responses shows that polynomials are closed under addition, we need to check the polynomial additions and ensure that they yield another polynomial in the correct form.

The expression is:

\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2)
\]

Let's combine the polynomials step by step:

1. Combine the \(x^2\) terms:
\[
0.923x^2 + 0.923x^2 = 1.846x^2
\]

2. Combine the \(x\) terms:
\[
-6.42x - 6.42x = -12.84x
\]

3. Combine the constant terms:
\[
2.98 + 2.98 = 5.96
\]

Putting these together, we have:
\[
1.846x^2 - 12.84x + 5.96
\]

Now we evaluate the provided responses:

1. **Response A:** \(3.903x^2 - 12.84x + 3.903\) - Incorrect.
2. **Response B:** \(10.994x^2 + 5.96\) - Incorrect.
3. **Response C:** \(1.846x^2 - 12.84x + 5.96\) - Correct.
4. **Response D:** \(2x^2 - 13x + 6\) - Incorrect.

The correct response that shows that polynomials are closed under addition is **Response C:**

\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2) = 1.846x^2 - 12.84x + 5.96
\]