find a polynomial function p(x) in standard form having leading coefficient 1, least possible degree, real coefficients, and the given zeros.

(5 + \sqrt{8}), (5 - \sqrt{8}), and 5

p(x) =

Question

find a polynomial function p(x) in standard form having leading coefficient 1, least possible degree, real coefficients, and the given zeros.

(5 + \sqrt{8}), (5 - \sqrt{8}), and 5

p(x) =

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Polynomial Zeros", "Standard Form Polynomials", "Irrational Conjugates" ], "current_concepts": [ "Polynomial Zeros", "Standard Form Polynomials", "Irrational Conjugates" ] } ## Set up the factored form of the polynomial $$ P(x) = (x - (5 + \sqrt{8}))(x - (5 - \sqrt{8}))(x - 5) $$ ## Expand the quadratic factor containing the irrational zeros $$ \begin{aligned} (x - (5 + \sqrt{8}))(x - (5 - \sqrt{8})) &= ((x - 5) - \sqrt{8})((x - 5) + \sqrt{8}) \ &= (x - 5)^2 - 8 \ &= x^2 - 10x + 25 - 8 \ &= x^2 - 10x + 17 \end{aligned} $$ ## Multiply by the remaining linear factor to find standard form $$ \begin{aligned} P(x) &= (x^2 - 10x + 17)(x - 5) \ &= x(x^2 - 10x + 17) - 5(x^2 - 10x + 17) \ &= x^3 - 10x^2 + 17x - 5x^2 + 50x - 85 \ &= x^3 - 15x^2 + 67x - 85 \end{aligned} $$ Find a polynomial function P(x) in standard form having leading coefficient 1, least possible degree, real coefficients, and the given zeros. $5 + \sqrt{8}, 5 - \sqrt{8}$, and 5 $P(x) =$ $x^3 - 15x^2 + 67x - 85$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Zeros" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the factored form of the polynomial

Expand the quadratic factor containing the irrational zeros

Multiply by the remaining linear factor to find standard form

Answer: