select the correct answer from each drop-down menu.

the graph of the function $y = x^2(x^2 - 6x + 9)$ has zeros of choose an answer, so the function has choose an answer distinct real zeros and choose an answer complex zeros.

Question

select the correct answer from each drop-down menu.

the graph of the function $y = x^2(x^2 - 6x + 9)$ has zeros of choose an answer, so the function has choose an answer distinct real zeros and choose an answer complex zeros.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Zeros of Polynomial Functions", "Multiplicity of Zeros" ], "new_concepts": [], "current_concepts": [ "Zeros of Polynomial Functions", "Multiplicity of Zeros", "Complex Roots of Polynomials" ] } ## Find the zeros of the function $$ \begin{aligned} y &= x^2(x^2 - 6x + 9) \ y &= x^2(x - 3)^2 \ x^2(x - 3)^2 &= 0 \implies x = 0 \text{ or } x = 3 \end{aligned} $$ ## Determine the multiplicities and distinct real zeros $$ \begin{aligned} &\text{Zero } x = 0 \text{ has multiplicity } 2 \ &\text{Zero } x = 3 \text{ has multiplicity } 2 \ &\text{Distinct real zeros: } 2 \quad (x = 0 \text{ and } x = 3) \end{aligned} $$ ## Determine the number of complex zeros $$ \begin{aligned} &\text{Degree of polynomial } y = x^4 - 6x^3 + 9x^2 \text{ is } 4 \ &\text{Total zeros (with multiplicity): } 4 \ &\text{All zeros } (0, 0, 3, 3) \text{ are real} \ &\text{Number of non-real complex zeros: } 0 \end{aligned} $$ The graph of the function $y = x^2(x^2 - 6x + 9)$ has zeros of $0\text{ and } 3$, so the function has $2$ distinct real zeros and $0$ complex zeros. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Zeros of Polynomial Functions" ] }

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Explanation:

Find the zeros of the function

Determine the multiplicities and distinct real zeros

Determine the number of complex zeros

Answer: