Rational Exprecei Algebra 2 Multiplying/Dividing Rational Expressions Name: \( \qquad \) Date: \( \qquad \) Period: \( \qquad \) Directions: Solve the expressions in the first column, then match the answer to the expression with the corresponding letter to solve the riddle. A. \( \frac{x^{2}-36}{-x^{2}-2 x+48} \cdot \frac{x+8}{x-3} \) \[ \frac{x-1}{7} \]

Sure, let's work through **Expression A** step by step to simplify it. The goal is to simplify the rational expression and verify if it matches the given answer \(\frac{x-1}{7}\). ### **Given Expression A:** \[ \frac{x^{2} - 36}{-x^{2} - 2x + 48} \cdot \frac{x + 8}{x - 3} \] ### **Step 1: Factor the Numerators and Denominators** 1. **Factor \(x^{2} - 36\):** \[ x^{2} - 36 = (x - 6)(x + 6) \] *(This is a difference of squares.)* 2. **Factor \(-x^{2} - 2x + 48\):** - First, factor out a \(-1\): \[ -x^{2} - 2x + 48 = - (x^{2} + 2x - 48) \] - Now, factor \(x^{2} + 2x - 48\): - We look for two numbers that multiply to \(-48\) and add to \(2\). These numbers are \(8\) and \(-6\). \[ x^{2} + 2x - 48 = (x + 8)(x - 6) \] - So, the denominator becomes: \[ - (x + 8)(x - 6) \] 3. **Factor \(x + 8\):** - This expression is already factored. 4. **Factor \(x - 3\):** - This expression is already factored. ### **Step 2: Rewrite the Original Expression with Factors** \[ \frac{(x - 6)(x + 6)}{- (x + 8)(x - 6)} \cdot \frac{x + 8}{x - 3} \] ### **Step 3: Simplify by Canceling Common Factors** 1. **Cancel \((x - 6)\) in the numerator and denominator:** \[ \frac{\cancel{(x - 6)}(x + 6)}{- (x + 8)\cancel{(x - 6)}} \cdot \frac{x + 8}{x - 3} = \frac{x + 6}{- (x + 8)} \cdot \frac{x + 8}{x - 3} \] 2. **Cancel \((x + 8)\) in the numerator and denominator:** \[ \frac{x + 6}{- \cancel{(x + 8)}} \cdot \frac{\cancel{(x + 8)}}{x - 3} = \frac{x + 6}{-1} \cdot \frac{1}{x - 3} \] ### **Step 4: Simplify the Expression Further** \[ \frac{x + 6}{-1} \cdot \frac{1}{x - 3} = -\frac{x + 6}{x - 3} \] ### **Final Simplified Form:** \[ -\frac{x + 6}{x - 3} \] ### **Comparison with Given Answer:** The simplified form of **Expression A** is: \[ -\frac{x + 6}{x - 3} \] However, the given answer was: \[ \frac{x - 1}{7} \] These two expressions are **not equivalent** unless specific values of \(x\) satisfy both, which generally they do not. This suggests one of the following: - There might have been a mistake in the provided expression or the given answer. - Additional expressions (e.g., B, C, etc.) and their corresponding answers are needed to complete the matching and solve the riddle as per the instructions. ### **Conclusion:** **Expression A** simplifies to \(-\frac{x + 6}{x - 3}\), which does not match the provided answer \(\frac{x - 1}{7}\). Please double-check the original expressions and answers for any discrepancies. If there are more expressions (like B, C, etc.), providing them would help in completing the matching process for the riddle.