select the correct answer.

which expression is equivalent to $\frac{(x - 1)^2}{x^2 - x - 12} \cdot \frac{x^2 + x - 6}{x^2 - 6x + 5}$ if no denominator equals zero?

$\bigcirc \frac{x^2 - 3x + 2}{x^2 - 9x + 20}$
$\bigcirc \frac{x^2 + x - 2}{x^2 - x - 20}$
$\bigcirc \frac{x^2 - 3x + 2}{x^2 - x - 20}$
$\bigcirc \frac{x^2 + 3x + 2}{x^2 + x - 20}$

Question

select the correct answer.

which expression is equivalent to $\frac{(x - 1)^2}{x^2 - x - 12} \cdot \frac{x^2 + x - 6}{x^2 - 6x + 5}$ if no denominator equals zero?

$\bigcirc \frac{x^2 - 3x + 2}{x^2 - 9x + 20}$
$\bigcirc \frac{x^2 + x - 2}{x^2 - x - 20}$
$\bigcirc \frac{x^2 - 3x + 2}{x^2 - x - 20}$
$\bigcirc \frac{x^2 + 3x + 2}{x^2 + x - 20}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Expression Division", "Polynomial Factoring", "Rational Expression Simplification" ], "new_concepts": [], "current_concepts": [ "Rational Expression Division", "Polynomial Factoring", "Rational Expression Simplification" ] } ## Factor each polynomial in the expression $$ \begin{aligned} (x-1)^2 &= (x-1)(x-1) \ x^2-x-12 &= (x-4)(x+3) \ x^2+x-6 &= (x+3)(x-2) \ x^2-6x+5 &= (x-5)(x-1) \end{aligned} $$ ## Rewrite division as multiplication and simplify $$ \begin{aligned} \frac{(x-1)^2}{x^2-x-12} \div \frac{x^2+x-6}{x^2-6x+5} &= \frac{(x-1)^2}{x^2-x-12} \cdot \frac{x^2-6x+5}{x^2+x-6} \ &= \frac{(x-1)(x-1)}{(x-4)(x+3)} \cdot \frac{(x-5)(x-1)}{(x+3)(x-2)} \ &= \frac{(x-1)^3(x-5)}{(x-4)(x+3)^2(x-2)} \end{aligned} $$ *Re-evaluating the operator in the image:* The operator is a multiplication dot ($\cdot$), not a division sign ($\div$). $$ \begin{aligned} \frac{(x-1)^2}{x^2-x-12} \cdot \frac{x^2+x-6}{x^2-6x+5} &= \frac{(x-1)(x-1)}{(x-4)(x+3)} \cdot \frac{(x+3)(x-2)}{(x-5)(x-1)} \ &= \frac{(x-1)(x-2)}{(x-4)(x-5)} \end{aligned} $$ ## Expand the simplified expression $$ \begin{aligned} \text{Numerator: } (x-1)(x-2) &= x^2 - 3x + 2 \ \text{Denominator: } (x-4)(x-5) &= x^2 - 9x + 20 \ \text{Equivalent Expression: } &\frac{x^2-3x+2}{x^2-9x+20} \end{aligned} $$

(A) $\frac{x^2-3x+2}{x^2-9x+20}$

(B) $\frac{x^2+x-2}{x^2-x-20}$

(C) $\frac{x^2-3x+2}{x^2-x-20}$

(D) $\frac{x^2+3x+2}{x^2+x-20}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Rational Expression Simplification" ] }

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QUESTION IMAGE

Question

Explanation:

Factor each polynomial in the expression

Rewrite division as multiplication and simplify

Expand the simplified expression

Answer: