Question

which statement is true about the graphed cubic function?

• the function has two distinct real zeros and one complex zero.
• the function has two distinct real zeros and no complex zeros.
• the function has one distinct real zero and two complex zeros.
• the function has three distinct real zeros and no complex zeros.

Explanation:

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"concepts_used": [
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"new_concepts": [
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</pre_analysis>

<reasoning>

Identify the real zeros from the graph

Using the Zeros of Polynomials knowledge point
\[

$$\begin{aligned} &\text{Real zeros correspond to the } x\text{-intercepts of the graph } y = f(x).\\ &\text{The graph crosses the } x\text{-axis at exactly one point: } x = 2.\\ &\text{Therefore, the function has exactly one distinct real zero.} \end{aligned}$$

\]

Determine the number of complex zeros

A cubic function is a polynomial of degree 3. By the Fundamental Theorem of Algebra, a degree 3 polynomial has exactly 3 zeros in the complex number system (counting multiplicity). Since there is exactly 1 real zero, the remaining 2 zeros must be non-real complex conjugate zeros. Thus, the function has one distinct real zero and two complex zeros.
</reasoning>

<answer>
<mcq-option>The function has two distinct real zeros and one complex zero.</mcq-option>
<mcq-option>The function has two distinct real zeros and no complex zeros.</mcq-option>
<mcq-correct>The function has one distinct real zero and two complex zeros.</mcq-correct>
<mcq-option>The function has three distinct real zeros and no complex zeros.</mcq-option>
</answer>

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

Answer:

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"Zeros of Polynomials",
"Complex Zeros of Polynomials",
"Fundamental Theorem of Algebra"
]
}
</pre_analysis>

<reasoning>

Identify the real zeros from the graph

Using the Zeros of Polynomials knowledge point
\[

$$\begin{aligned} &\text{Real zeros correspond to the } x\text{-intercepts of the graph } y = f(x).\\ &\text{The graph crosses the } x\text{-axis at exactly one point: } x = 2.\\ &\text{Therefore, the function has exactly one distinct real zero.} \end{aligned}$$

\]

Determine the number of complex zeros

A cubic function is a polynomial of degree 3. By the Fundamental Theorem of Algebra, a degree 3 polynomial has exactly 3 zeros in the complex number system (counting multiplicity). Since there is exactly 1 real zero, the remaining 2 zeros must be non-real complex conjugate zeros. Thus, the function has one distinct real zero and two complex zeros.
</reasoning>

<answer>
<mcq-option>The function has two distinct real zeros and one complex zero.</mcq-option>
<mcq-option>The function has two distinct real zeros and no complex zeros.</mcq-option>
<mcq-correct>The function has one distinct real zero and two complex zeros.</mcq-correct>
<mcq-option>The function has three distinct real zeros and no complex zeros.</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
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"Mathematics",
"Algebra",
"Zeros of Polynomials"
]
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</post_analysis>