Question

the following function is given.

\\f(x) = x^3 - 3x^2 - 4x + 12\\

a. list all rational zeros that are possible according to the rational zero theorem.

1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12
(use a comma to separate answers as needed.)

b. use synthetic division to test several possible rational zeros in order to identify one actual zero.

one rational zero of the given function is .
(simplify your answer.)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Rational Root Theorem",
"Synthetic Division"
],
"new_concepts": [],
"current_concepts": [
"Rational Root Theorem",
"Synthetic Division",
"Finding Zeros of Factored Polynomials"
]
}
</pre_analysis>

<reasoning>

Identify the given function and target

We are given the cubic polynomial function:
\[f(x) = x^3 - 3x^2 - 4x + 12\]
Our goal is to find one actual rational zero of this function using synthetic division to test the candidates listed in part (a).

Test candidates using synthetic division

Using the Synthetic Division and Rational Root Theorem knowledge points
\[

$$\begin{array}{r|rrrr} 2 & 1 & -3 & -4 & 12 \\ & & 2 & -2 & -12 \\ \hline & 1 & -1 & -6 & 0 \end{array}$$

\]
Since the remainder is \(0\), \(x = 2\) is an actual rational zero of the function.

Verify the zero by direct substitution

Using the Finding Zeros of Factored Polynomials knowledge point
\[
f(2) = 2^3 - 3(2)^2 - 4(2) + 12 = 8 - 12 - 8 + 12 = 0
\]
</reasoning>

<answer>
One rational zero of the given function is <blank>2</blank>
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Rational Root Theorem",
"Synthetic Division"
],
"new_concepts": [],
"current_concepts": [
"Rational Root Theorem",
"Synthetic Division",
"Finding Zeros of Factored Polynomials"
]
}
</pre_analysis>

<reasoning>

Identify the given function and target

We are given the cubic polynomial function:
\[f(x) = x^3 - 3x^2 - 4x + 12\]
Our goal is to find one actual rational zero of this function using synthetic division to test the candidates listed in part (a).

Test candidates using synthetic division

Using the Synthetic Division and Rational Root Theorem knowledge points
\[

$$\begin{array}{r|rrrr} 2 & 1 & -3 & -4 & 12 \\ & & 2 & -2 & -12 \\ \hline & 1 & -1 & -6 & 0 \end{array}$$

\]
Since the remainder is \(0\), \(x = 2\) is an actual rational zero of the function.

Verify the zero by direct substitution

Using the Finding Zeros of Factored Polynomials knowledge point
\[
f(2) = 2^3 - 3(2)^2 - 4(2) + 12 = 8 - 12 - 8 + 12 = 0
\]
</reasoning>

<answer>
One rational zero of the given function is <blank>2</blank>
</answer>

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