which two expressions are prime polynomials?

$x^2 - 5x + 3$

$x^4 - xy^3$

$x^2 - 7x + 6$

$2xy - 4x^2y^3 + 5x^3 - 8y^2$

$x^4y^3 - 3x^3 + 4x^2y^2 - x$

Question

which two expressions are prime polynomials?

$x^2 - 5x + 3$

$x^4 - xy^3$

$x^2 - 7x + 6$

$2xy - 4x^2y^3 + 5x^3 - 8y^2$

$x^4y^3 - 3x^3 + 4x^2y^2 - x$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Factoring Polynomials", "Factoring Trinomials" ], "new_concepts": [ "Prime Polynomials" ], "current_concepts": [ "Factoring Polynomials", "Factoring Trinomials", "Prime Polynomials" ] } ## Analyze the quadratic trinomials $$ \begin{aligned} &x^2 - 5x + 3 \implies \text{No integers } p, q \text{ satisfy } p+q = -5 \text{ and } pq = 3 \implies \text{Prime} \ &x^2 - 7x + 6 = (x-6)(x-1) \implies \text{Not prime} \end{aligned} $$ ## Analyze the binomial and four-term expressions $$ \begin{aligned} &x^4 - xy^3 = x(x^3 - y^3) = x(x-y)(x^2+xy+y^2) \implies \text{Not prime} \ &2xy - 4x^2y^3 + 5x^3 - 8y^2 \implies \text{No common factors or successful grouping} \implies \text{Prime} \ &x^4y^3 - 3x^3 + 4x^2y^2 - x = x(x^3y^3 - 3x^2 + 4xy^2 - 1) \implies \text{Not prime} \end{aligned} $$ ## Identify the two prime polynomials $$ \begin{aligned} &\text{Prime 1: } x^2 - 5x + 3 \ &\text{Prime 2: } 2xy - 4x^2y^3 + 5x^3 - 8y^2 \end{aligned} $$

[x] $x^2 - 5x + 3$

[ ] $x^4 - xy^3$

[ ] $x^2 - 7x + 6$

[x] $2xy - 4x^2y^3 + 5x^3 - 8y^2$

[ ] $x^4y^3 - 3x^3 + 4x^2y^2 - x$

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

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QUESTION IMAGE

Question

Explanation:

Analyze the quadratic trinomials

Analyze the binomial and four-term expressions

Identify the two prime polynomials

Answer: