simplify: \frac{7z + 5}{z^2 - 1} - \frac{-2z^2 + 7z + 7}{-z^2 + 1}.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Difference of Squares", "Subtracting Rational Expressions" ], "new_concepts": [], "current_concepts": [ "Difference of Squares", "Subtracting Rational Expressions", "Simplifying Rational Expressions" ] } ## Analyze the denominators Using the *Difference of Squares* knowledge point $$ \begin{aligned} z^2 - 1 &= (z - 1)(z + 1)\ -z^2 + 1 &= -(z^2 - 1) = -(z - 1)(z + 1) \end{aligned} $$ ## Rewrite the subtraction with a common denominator Using the *Subtracting Rational Expressions* knowledge point $$ \begin{aligned} \frac{7z + 5}{z^2 - 1} - \frac{-2z^2 + 7z + 7}{-z^2 + 1} &= \frac{7z + 5}{z^2 - 1} - \frac{-2z^2 + 7z + 7}{-(z^2 - 1)}\ &= \frac{7z + 5}{z^2 - 1} + \frac{-2z^2 + 7z + 7}{z^2 - 1} \end{aligned} $$ ## Combine the numerators Using the *Subtracting Rational Expressions* knowledge point $$ \begin{aligned} \frac{(7z + 5) + (-2z^2 + 7z + 7)}{z^2 - 1} &= \frac{-2z^2 + 14z + 12}{z^2 - 1} \end{aligned} $$ ## Factor and simplify the expression Using the *Difference of Squares* knowledge point $$ \begin{aligned} \frac{-2(z^2 - 7z - 6)}{(z - 1)(z + 1)} \end{aligned} $$ $\frac{-2z^2 + 14z + 12}{z^2 - 1}$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Subtracting Rational Expressions" ] }

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Question

Explanation:

Analyze the denominators

Rewrite the subtraction with a common denominator

Combine the numerators

Factor and simplify the expression

Answer: