select the correct answer.

which expression is a prime polynomial?

$10x^4 - 5x^3 + 70x^2 + 3x$
$x^4 + 20x^2 - 100$
$x^3 - 27y^6$
$3x^2 + 18y$

Question

select the correct answer.

which expression is a prime polynomial?

$10x^4 - 5x^3 + 70x^2 + 3x$
$x^4 + 20x^2 - 100$
$x^3 - 27y^6$
$3x^2 + 18y$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Factoring", "Difference of Cubes" ], "new_concepts": [ "Prime Polynomial" ], "current_concepts": [ "Polynomial Factoring", "Difference of Cubes", "Prime Polynomial" ] } ## Analyze factorability of each option $$ \begin{aligned} &10x^4 - 5x^3 + 70x^2 + 3x = x(10x^3 - 5x^2 + 70x + 3) \quad (\text{Factorable by GCF}) \ &x^4 + 20x^2 - 100 \quad (\text{Cannot be factored over the integers; discriminant of } u^2 + 20u - 100 \text{ is } 20^2 - 4(1)(-100) = 800, \text{ not a perfect square}) \ &x^3 - 27y^6 = (x)^3 - (3y^2)^3 = (x - 3y^2)(x^2 + 3xy^2 + 9y^4) \quad (\text{Factorable by difference of cubes}) \ &3x^2 + 18y = 3(x^2 + 6y) \quad (\text{Factorable by GCF}) \end{aligned} $$ ## Identify the prime polynomial $$ x^4 + 20x^2 - 100 \text{ is prime because it cannot be factored into lower-degree polynomials with integer coefficients.} $$

(A) $10x^4 - 5x^3 + 70x^2 + 3x$

(B) $x^4 + 20x^2 - 100$

(C) $x^3 - 27y^6$

(D) $3x^2 + 18y$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Factoring" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze factorability of each option

Identify the prime polynomial

Answer: