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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Complex Conjugate Theorem", "Polynomial Multiplicity" ], "current_concepts": [ "Polynomial Zeros", "Standard Form Polynomials", "Complex Conjugate Theorem", "Polynomial Multiplicity" ] } ## Identify the zeros and their multiplicities $$ \begin{aligned} &x_1 = 1 \quad (\text{multiplicity } 2) \ &x_2 = 3 - 5i \quad (\text{multiplicity } 1) \ &x_3 = 3 + 5i \quad (\text{multiplicity } 1 \text{ by Complex Conjugate Theorem}) \end{aligned} $$ ## Write the polynomial in factored form $$ \begin{aligned} P(x) &= 1 \cdot (x - 1)^2 (x - (3 - 5i))(x - (3 + 5i)) \ &= (x^2 - 2x + 1)((x - 3) + 5i)((x - 3) - 5i) \ &= (x^2 - 2x + 1)((x - 3)^2 - (5i)^2) \ &= (x^2 - 2x + 1)(x^2 - 6x + 9 + 25) \ &= (x^2 - 2x + 1)(x^2 - 6x + 34) \end{aligned} $$ ## Expand to standard form $$ \begin{aligned} P(x) &= x^2(x^2 - 6x + 34) - 2x(x^2 - 6x + 34) + 1(x^2 - 6x + 34) \ &= x^4 - 6x^3 + 34x^2 - 2x^3 + 12x^2 - 68x + x^2 - 6x + 34 \ &= x^4 - 8x^3 + 47x^2 - 74x + 34 \end{aligned} $$ $P(x) = x^4 - 8x^3 + 47x^2 - 74x + 34$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Standard Form Polynomials" ] }

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Identify the zeros and their multiplicities

Write the polynomial in factored form

Expand to standard form

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