the graph of a quadratic function intercepts the $x$-axis in two places and the $y$-axis in one place.
according to the fundamental theorem of algebra, which of the following statements is correct?

- the quadratic function has two distinct real zeros and one distinct complex zero.
- the quadratic function has two distinct real zeros.
- the quadratic function has one distinct real zero and one distinct complex zero.
- the quadratic function has no real zeros and two complex zeros.

Question

the graph of a quadratic function intercepts the $x$-axis in two places and the $y$-axis in one place.
according to the fundamental theorem of algebra, which of the following statements is correct?

- the quadratic function has two distinct real zeros and one distinct complex zero.
- the quadratic function has two distinct real zeros.
- the quadratic function has one distinct real zero and one distinct complex zero.
- the quadratic function has no real zeros and two complex zeros.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Fundamental Theorem of Algebra", "Quadratic Zeros" ], "current_concepts": [ "Fundamental Theorem of Algebra", "Quadratic Zeros", "Complex Conjugate Root Theorem" ] } ## Identify the degree of the function and the number of zeros $$ \begin{aligned} &\text{Degree of quadratic function } n = 2\ &\text{Total complex zeros (including multiplicity)} = 2 \end{aligned} $$ ## Analyze the given x-intercepts $$ \begin{aligned} &\text{Number of } x\text{-intercepts} = 2\ &\text{Number of distinct real zeros} = 2 \end{aligned} $$ ## Determine the remaining zeros $$ \begin{aligned} &\text{Remaining zeros} = \text{Total zeros} - \text{Real zeros} = 2 - 2 = 0\ &\text{Number of non-real complex zeros} = 0 \end{aligned} $$

The quadratic function has two distinct real zeros and one distinct complex zero.

The quadratic function has two distinct real zeros.

The quadratic function has one distinct real zero and one distinct complex zero.

The quadratic function has no real zeros and two complex zeros.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Fundamental Theorem of Algebra" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the degree of the function and the number of zeros

Analyze the given x-intercepts

Determine the remaining zeros

Answer: