Question

finding the factors of a trinomial

what are the factors of \\(x^2 - 4x - 5\\)? check all that apply.

\\(x - 5\\)
\\(x - 2\\)
\\(x - 1\\)
\\(x + 1\\)
\\(x + 2\\)
\\(x + 5\\)

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": [
"Factoring Trinomials"
],
"current_concepts": [
"Factoring Trinomials",
"Algebra Tiles"
]
}
</pre_analysis>

<reasoning>

Identify the trinomial to factor

The given trinomial is:
\[
x^2 - 4x - 5
\]

Find two numbers that multiply to \(ac\) and add to \(b\)

For the quadratic expression \(ax^2 + bx + c\), we have \(a = 1\), \(b = -4\), and \(c = -5\).
We need two integers whose product is \(a \cdot c = -5\) and whose sum is \(b = -4\).
The factors of \(-5\) are:
\[

$$\begin{aligned} &(-5) \cdot 1 = -5 \quad \text{and} \quad -5 + 1 = -4\\ &5 \cdot (-1) = -5 \quad \text{and} \quad 5 + (-1) = 4 \end{aligned}$$

\]
The correct pair is \(-5\) and \(1\).

Write the factored form

Using the found integers, we write the trinomial in its factored form:
\[
x^2 - 4x - 5 = (x - 5)(x + 1)
\]
Thus, the factors are \(x - 5\) and \(x + 1\).
</reasoning>

<answer>
<mcq-correct>(A) \(x - 5\)</mcq-correct>
<mcq-option>(B) \(x - 2\)</mcq-option>
<mcq-option>(C) \(x - 1\)</mcq-option>
<mcq-correct>(D) \(x + 1\)</mcq-correct>
<mcq-option>(E) \(x + 2\)</mcq-option>
<mcq-option>(F) \(x + 5\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Factoring Trinomials"
]
}
</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": [
"Factoring Trinomials"
],
"current_concepts": [
"Factoring Trinomials",
"Algebra Tiles"
]
}
</pre_analysis>

<reasoning>

Identify the trinomial to factor

The given trinomial is:
\[
x^2 - 4x - 5
\]

Find two numbers that multiply to \(ac\) and add to \(b\)

For the quadratic expression \(ax^2 + bx + c\), we have \(a = 1\), \(b = -4\), and \(c = -5\).
We need two integers whose product is \(a \cdot c = -5\) and whose sum is \(b = -4\).
The factors of \(-5\) are:
\[

$$\begin{aligned} &(-5) \cdot 1 = -5 \quad \text{and} \quad -5 + 1 = -4\\ &5 \cdot (-1) = -5 \quad \text{and} \quad 5 + (-1) = 4 \end{aligned}$$

\]
The correct pair is \(-5\) and \(1\).

Write the factored form

Using the found integers, we write the trinomial in its factored form:
\[
x^2 - 4x - 5 = (x - 5)(x + 1)
\]
Thus, the factors are \(x - 5\) and \(x + 1\).
</reasoning>

<answer>
<mcq-correct>(A) \(x - 5\)</mcq-correct>
<mcq-option>(B) \(x - 2\)</mcq-option>
<mcq-option>(C) \(x - 1\)</mcq-option>
<mcq-correct>(D) \(x + 1\)</mcq-correct>
<mcq-option>(E) \(x + 2\)</mcq-option>
<mcq-option>(F) \(x + 5\)</mcq-option>
</answer>

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