Question

the sum of two polynomials is (10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2). if one addend is (-5a^2b^2 + 12a^2b - 5), what is the other addend?

(15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7)
(5a^2b^2 - 20a^2b^2 + 7)
(5a^2b^2 + 4a^2b^2 + 6ab - 4ab - 3)
(-15a^2b^2 + 20a^2b^2 - 6ab + 4ab - 7)

Explanation:

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<reasoning>

Set up the subtraction equation

\[
P = (10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2) - (-5a^2b^2 + 12a^2b - 5)
\]

Distribute the negative sign

\[
P = 10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2 + 5a^2b^2 - 12a^2b + 5
\]

Combine like terms

\[

$$\begin{aligned} P &= (10a^2b^2 + 5a^2b^2) + (-8a^2b - 12a^2b) + 6ab^2 - 4ab + (2 + 5) \\ P &= 15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7\)</mcq-correct>
<mcq-option>(B) \(5a^2b^2 - 20a^2b^2 + 7\)</mcq-option>
<mcq-option>(C) \(5a^2b^2 + 4a^2b^2 + 6ab - 4ab - 3\)</mcq-option>
<mcq-option>(D) \(-15a^2b^2 + 20a^2b^2 - 6ab + 4ab - 7\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Subtraction"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
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"concepts_used": [
"Polynomial Subtraction",
"Like Terms"
],
"new_concepts": [],
"current_concepts": [
"Polynomial Subtraction",
"Like Terms",
"Polynomial Addition"
]
}
</pre_analysis>

<reasoning>

Set up the subtraction equation

\[
P = (10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2) - (-5a^2b^2 + 12a^2b - 5)
\]

Distribute the negative sign

\[
P = 10a^2b^2 - 8a^2b + 6ab^2 - 4ab + 2 + 5a^2b^2 - 12a^2b + 5
\]

Combine like terms

\[

$$\begin{aligned} P &= (10a^2b^2 + 5a^2b^2) + (-8a^2b - 12a^2b) + 6ab^2 - 4ab + (2 + 5) \\ P &= 15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(15a^2b^2 - 20a^2b + 6ab^2 - 4ab + 7\)</mcq-correct>
<mcq-option>(B) \(5a^2b^2 - 20a^2b^2 + 7\)</mcq-option>
<mcq-option>(C) \(5a^2b^2 + 4a^2b^2 + 6ab - 4ab - 3\)</mcq-option>
<mcq-option>(D) \(-15a^2b^2 + 20a^2b^2 - 6ab + 4ab - 7\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Subtraction"
]
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