let $x^3 - 2x^2 - 6x + 7 = 0$ be a polynomial equation.

a) find all potential rational solution(s) using the rational root theorem. separate multiple solutions with commas if necessary.

b) find all the distinct, actual rational solution(s). separate multiple solutions with commas if necessary. if no solutions exist, enter none.

Question

let $x^3 - 2x^2 - 6x + 7 = 0$ be a polynomial equation.

a) find all potential rational solution(s) using the rational root theorem. separate multiple solutions with commas if necessary.

b) find all the distinct, actual rational solution(s). separate multiple solutions with commas if necessary. if no solutions exist, enter none.

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Rational Root Theorem", "Polynomial Roots" ], "current_concepts": [ "Rational Root Theorem", "Polynomial Roots" ] } ## Identify potential rational roots $$ \begin{aligned} &\text{Equation: } x^3 - 2x^2 - 6x + 7 = 0 \ &\text{Leading coefficient } a_n = 1 \implies \text{factors of } 1: \pm 1 \ &\text{Constant term } a_0 = 7 \implies \text{factors of } 7: \pm 1, \pm 7 \ &\text{Potential rational roots } \frac{p}{q}: \pm 1, \pm 7 \end{aligned} $$ ## Test potential roots to find actual rational solutions $$ \begin{aligned} &P(1) = 1^3 - 2(1)^2 - 6(1) + 7 = 1 - 2 - 6 + 7 = 0 \implies x = 1 \text{ is a root.} \ &P(-1) = (-1)^3 - 2(-1)^2 - 6(-1) + 7 = -1 - 2 + 6 + 7 = 10 e 0 \ &P(7) = 7^3 - 2(7)^2 - 6(7) + 7 = 343 - 98 - 42 + 7 = 210 e 0 \ &P(-7) = (-7)^3 - 2(-7)^2 - 6(-7) + 7 = -343 - 98 + 42 + 7 = -392 e 0 \end{aligned} $$ ## Find remaining roots $$ \begin{aligned} &(x^3 - 2x^2 - 6x + 7) \div (x - 1) = x^2 - x - 7 \ &x^2 - x - 7 = 0 \implies x = \frac{1 \pm \sqrt{1 - 4(1)(-7)}}{2} = \frac{1 \pm \sqrt{29}}{2} \quad (\text{irrational}) \end{aligned} $$ ## Question a -7,-1,1,7 ## Question b 1 { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Rational Root Theorem" ] }

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Question

Explanation:

Identify potential rational roots

Test potential roots to find actual rational solutions

Find remaining roots

Answer:

Question a

Question b