the variables $a$, $b$, and $c$ represent polynomials where $a = x^2$, $b = 3x + 2$, and $c = x - 3$. what is $ab - c^2$ in simplest form?

$3x^3 + 2x^2 - x + 3$
$3x^3 + 2x^2 - x - 3$
$3x^3 + x^2 - 6x + 9$
$3x^3 + x^2 + 6x - 9$

Question

the variables $a$, $b$, and $c$ represent polynomials where $a = x^2$, $b = 3x + 2$, and $c = x - 3$. what is $ab - c^2$ in simplest form?

$3x^3 + 2x^2 - x + 3$
$3x^3 + 2x^2 - x - 3$
$3x^3 + x^2 - 6x + 9$
$3x^3 + x^2 + 6x - 9$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Multiplication", "Polynomial Subtraction", "Perfect Square Trinomial" ], "new_concepts": [], "current_concepts": [ "Polynomial Multiplication", "Polynomial Subtraction", "Perfect Square Trinomial", "Polynomial Operations" ] } ## Substitute the given polynomials $$ \begin{aligned} A &= x^2 \ B &= 3x + 2 \ C &= x - 3 \ AB - C^2 &= (x^2)(3x + 2) - (x - 3)^2 \end{aligned} $$ ## Expand each term $$ \begin{aligned} AB &= x^2(3x + 2) = 3x^3 + 2x^2 \ C^2 &= (x - 3)^2 = x^2 - 6x + 9 \end{aligned} $$ ## Subtract the expanded expressions $$ \begin{aligned} AB - C^2 &= (3x^3 + 2x^2) - (x^2 - 6x + 9) \ &= 3x^3 + 2x^2 - x^2 + 6x - 9 \ &= 3x^3 + x^2 + 6x - 9 \end{aligned} $$

(A) $3x^3 + 2x^2 - x + 3$

(B) $3x^3 + 2x^2 - x - 3$

(C) $3x^3 + x^2 - 6x + 9$

(D) $3x^3 + x^2 + 6x - 9$

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

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QUESTION IMAGE

Question

Explanation:

Substitute the given polynomials

Expand each term

Subtract the expanded expressions

Answer: