select the correct answer.

which expression is equivalent to $\frac{x + 3}{x^2 - 2x - 3} \div \frac{x^2 + 2x - 3}{x + 1}$ if no denominator equals zero?

$\bigcirc \frac{1}{x^2 - 2x - 3}$
$\bigcirc \frac{1}{x^2 - 4x + 3}$
$\bigcirc \frac{1}{x^2 + 2x - 3}$
$\bigcirc \frac{x + 3}{x + 1}$

Question

select the correct answer.

which expression is equivalent to $\frac{x + 3}{x^2 - 2x - 3} \div \frac{x^2 + 2x - 3}{x + 1}$ if no denominator equals zero?

$\bigcirc \frac{1}{x^2 - 2x - 3}$
$\bigcirc \frac{1}{x^2 - 4x + 3}$
$\bigcirc \frac{1}{x^2 + 2x - 3}$
$\bigcirc \frac{x + 3}{x + 1}$

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", "Rational Expression Simplification" ], "new_concepts": [], "current_concepts": [ "Rational Expression Division", "Rational Expression Simplification" ] } ## Convert division to multiplication by the reciprocal $$ \frac{x+3}{x^2-2x-3} \div \frac{x^2+2x-3}{x+1} = \frac{x+3}{x^2-2x-3} \cdot \frac{x+1}{x^2+2x-3} $$ ## Factor all quadratic trinomials $$ \begin{aligned} x^2-2x-3 &= (x-3)(x+1) \ x^2+2x-3 &= (x+3)(x-1) \end{aligned} $$ ## Substitute factors and simplify the expression $$ \begin{aligned} \frac{x+3}{(x-3)(x+1)} \cdot \frac{x+1}{(x+3)(x-1)} &= \frac{(x+3)(x+1)}{(x-3)(x+1)(x+3)(x-1)} \ &= \frac{1}{(x-3)(x-1)} \ &= \frac{1}{x^2-4x+3} \end{aligned} $$

(A) $\frac{1}{x^2-2x-3}$

(B) $\frac{1}{x^2-4x+3}$

(C) $\frac{1}{x^2+2x-3}$

(D) $\frac{x+3}{x+1}$

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

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QUESTION IMAGE

Question

Explanation:

Convert division to multiplication by the reciprocal

Factor all quadratic trinomials

Substitute factors and simplify the expression

Answer: