Question

factor the polynomial completely using the x method.
\\x^2 + 13x - 48\\

what is the four-term polynomial and factored form of the polynomial?

\\(x^2 + 7x + 6x - 48 = (x + 7)(x + 6)\\)
\\(x^2 - 12x + 4x - 48 = (x - 12)(x + 4)\\)
\\(x^2 + 16x - 3x - 48 = (x + 16)(x - 3)\\)
\\(x^2 - 16x + 3x - 48 = (x - 16)(x + 3)\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"X Method Factoring",
"Factoring Trinomials"
]
}
</pre_analysis>

<reasoning>

Identify coefficients

\[
a = 1, \quad b = 13, \quad c = -48
\]
\[
ac = -48
\]

Find factors of ac that sum to b

\[
16 \times (-3) = -48
\]
\[
16 + (-3) = 13
\]

Rewrite trinomial as four-term polynomial and factor

\[
x^2 + 16x - 3x - 48 = x(x + 16) - 3(x + 16) = (x + 16)(x - 3)
\]
</reasoning>

<answer>
<mcq-option>(A) \(x^2 + 7x + 6x - 48 = (x + 7)(x + 6)\)</mcq-option>
<mcq-option>(B) \(x^2 - 12x + 4x - 48 = (x - 12)(x + 4)\)</mcq-option>
<mcq-correct>(C) \(x^2 + 16x - 3x - 48 = (x + 16)(x - 3)\)</mcq-correct>
<mcq-option>(D) \(x^2 - 16x + 3x - 48 = (x - 16)(x + 3)\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Factoring Polynomials"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"X Method Factoring",
"Factoring Trinomials"
]
}
</pre_analysis>

<reasoning>

Identify coefficients

\[
a = 1, \quad b = 13, \quad c = -48
\]
\[
ac = -48
\]

Find factors of ac that sum to b

\[
16 \times (-3) = -48
\]
\[
16 + (-3) = 13
\]

Rewrite trinomial as four-term polynomial and factor

\[
x^2 + 16x - 3x - 48 = x(x + 16) - 3(x + 16) = (x + 16)(x - 3)
\]
</reasoning>

<answer>
<mcq-option>(A) \(x^2 + 7x + 6x - 48 = (x + 7)(x + 6)\)</mcq-option>
<mcq-option>(B) \(x^2 - 12x + 4x - 48 = (x - 12)(x + 4)\)</mcq-option>
<mcq-correct>(C) \(x^2 + 16x - 3x - 48 = (x + 16)(x - 3)\)</mcq-correct>
<mcq-option>(D) \(x^2 - 16x + 3x - 48 = (x - 16)(x + 3)\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Factoring Polynomials"
]
}
</post_analysis>