find the polynomial in standard form of lowest degree having leading coefficient 1, real coefficients with a zero of 1 (multiplicity 2), and zero $3 - 5i$.

$p(x) =$ (simplify your answer.)

Question

find the polynomial in standard form of lowest degree having leading coefficient 1, real coefficients with a zero of 1 (multiplicity 2), and zero $3 - 5i$.

$p(x) =$   (simplify your answer.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Standard Form Polynomials", "Complex Conjugate Theorem", "Polynomial Multiplicity" ], "new_concepts": [], "current_concepts": [ "Standard Form Polynomials", "Complex Conjugate Theorem", "Polynomial Multiplicity" ] } ## Identify all 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 factored form of the polynomial $$ \begin{aligned} P(x) &= (x - 1)^2 [x - (3 - 5i)][x - (3 + 5i)] \ &= (x^2 - 2x + 1)[(x - 3)^2 - (5i)^2] \ &= (x^2 - 2x + 1)(x^2 - 6x + 9 + 25) \ &= (x^2 - 2x + 1)(x^2 - 6x + 34) \end{aligned} $$ ## Expand to standard form $$ \begin{aligned} P(x) &= x^2(x^2 - 6x + 34) - 2x(x^2 - 6x + 34) + 1(x^2 - 6x + 34) \ &= (x^4 - 6x^3 + 34x^2) - (2x^3 - 12x^2 + 68x) + (x^2 - 6x + 34) \ &= x^4 - 8x^3 + 47x^2 - 74x + 34 \end{aligned} $$ Find the polynomial in standard form of lowest degree having leading coefficient 1, real coefficients with a zero of 1 (multiplicity 2), and zero $3 - 5i$. $P(x) =$ $x^4 - 8x^3 + 47x^2 - 74x + 34$ (Simplify your answer.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Standard Form Polynomials" ] }

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Question

Explanation:

Identify all zeros and their multiplicities

Write the factored form of the polynomial

Expand to standard form

Answer: