Question

which property does each equation demonstrate?
\\(x^2 + 2x = 2x + x^2\\)

\\((3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3\\)

\\((2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x)\\)

associative property
closure property
commutative property

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 3,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Algebraic Properties",
"Closure Property",
"Commutative Property"
],
"current_concepts": [
"Algebraic Properties",
"Closure Property",
"Commutative Property",
"Associative Property"
]
}
</pre_analysis>

<reasoning>

Analyze the first equation

\[
x^2 + 2x = 2x + x^2
\]
\[
A + B = B + A
\]

Analyze the second equation

\[
(3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3
\]
\[
P_1 - P_2 = P_3
\]

Analyze the third equation

\[
(2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x)
\]
\[
A + B = B + A
\]
</reasoning>

<answer>

Question 1

For \(x^2 + 2x = 2x + x^2\):
<blank>commutative property</blank>

Question 2

For \((3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3\):
<blank>closure property</blank>

Question 3

For \((2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x)\):
<blank>commutative property</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Algebraic Properties"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 3,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Algebraic Properties",
"Closure Property",
"Commutative Property"
],
"current_concepts": [
"Algebraic Properties",
"Closure Property",
"Commutative Property",
"Associative Property"
]
}
</pre_analysis>

<reasoning>

Analyze the first equation

\[
x^2 + 2x = 2x + x^2
\]
\[
A + B = B + A
\]

Analyze the second equation

\[
(3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3
\]
\[
P_1 - P_2 = P_3
\]

Analyze the third equation

\[
(2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x)
\]
\[
A + B = B + A
\]
</reasoning>

<answer>

Question 1

For \(x^2 + 2x = 2x + x^2\):
<blank>commutative property</blank>

Question 2

For \((3z^4 + 2z^3) - (2z^4 + z^3) = z^4 + z^3\):
<blank>closure property</blank>

Question 3

For \((2x^2 + 7x) + (2y^2 + 6y) = (2y^2 + 6y) + (2x^2 + 7x)\):
<blank>commutative property</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Algebraic Properties"
]
}
</post_analysis>