Question

Which of the following shows that polynomials are a closed system under addition

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=3.903x2−12.84x+3.903

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

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

(0.923x2−6.42x+2.98)+(2.98−6.42x+0.923x2)=10.994x2+5.96

Answer 1

what is an answer?

Answer 2

To determine which of the given options correctly shows that polynomials are a closed system under addition, we need to perform the addition on the left-hand side for each option and compare it to the result on the right-hand side.

Let's evaluate the addition of the two polynomials in each option:

1. **Option 1:**
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2)
\]
Combine like terms:
- For \(x^2\): \(0.923x^2 + 0.923x^2 = 1.846x^2\)
- For \(x\): \(-6.42x - 6.42x = -12.84x\)
- Constant: \(2.98 + 2.98 = 5.96\)
Resulting polynomial:
\[
1.846x^2 - 12.84x + 5.96
\]
So this does not match \(3.903x^2 - 12.84x + 3.903\).

2. **Option 2:**
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2)
\]
We already did this and found the result to be
\[
1.846x^2 - 12.84x + 5.96
\]
So this does not match \(1.846x^2 - 12.84x + 5.96\).

3. **Option 3:**
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2)
\]
We already evaluated this, and it equals
\[
1.846x^2 - 12.84x + 5.96
\]
So this does not match \(2x^2 - 13x + 6\).

4. **Option 4:**
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2)
\]
As established, the result is
\[
1.846x^2 - 12.84x + 5.96
\]
So this does not match \(10.994x^2 + 5.96\).

After evaluating all options, none of the provided options show the addition correctly that would confirm polynomials are closed under addition, because the computations lead to \(1.846x^2 - 12.84x + 5.96\) consistently across all cases, but none of the right-hand results corresponded to that result correctly. As such, none of the given options are correct reflections of polynomial addition leading to a valid polynomial expression.

Answer 3

None of the given options correctly demonstrate that polynomials are a closed system under addition because they all yield a result of \(1.846x^2 - 12.84x + 5.96\), while the right-hand sides do not match this result.

Here’s the breakdown again for clarity:

1. **Sum of the polynomials:**
\[
(0.923x^2 - 6.42x + 2.98) + (2.98 - 6.42x + 0.923x^2)
= (0.923 + 0.923)x^2 + (-6.42 - 6.42)x + (2.98 + 2.98)
= 1.846x^2 - 12.84x + 5.96
\]

None of the options provided match \(1.846x^2 - 12.84x + 5.96\).

So, **the answer is: none of the options correctly shows that polynomials are a closed system under addition.**