To simplify the polynomial expression \(\frac{x^2 - 1}{x + 1} \cdot \frac{x + 5}{x - 5}\), we can start breaking it down step by step.
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Factor \(x^2 - 1\): \[ x^2 - 1 = (x - 1)(x + 1) \]
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Rewrite the given expression: \[ \frac{(x - 1)(x + 1)}{x + 1} \cdot \frac{x + 5}{x - 5} \]
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Cancel the common \(x + 1\) term: \[ (x - 1) \cdot \frac{x + 5}{x - 5} \]
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Now multiply: \[ \frac{(x - 1)(x + 5)}{x - 5} \]
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Distributing in the numerator: \[ (x - 1)(x + 5) = x^2 + 5x - x - 5 = x^2 + 4x - 5 \]
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Therefore, the expression is now: \[ \frac{x^2 + 4x - 5}{x - 5} \]
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Now we can check if \(x^2 + 4x - 5\) can be factored further. We find its roots by factoring or using the quadratic formula: \[ x^2 + 4x - 5 = (x + 5)(x - 1) \]
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Substitute back: \[ \frac{(x + 5)(x - 1)}{x - 5} \]
Now, the expression is simplified as much as possible, and we find that it is:
\[ \frac{(x + 5)(x - 1)}{x - 5} \]
This expression can represent various values depending on \(x\), but does not directly match any of the responses directly as simplifications.
Assuming we were meant to evaluate this polynomial for some specific values of \(x\), none of the answer choices seem fully accurate based on the complete transformation presented here.
Please review your options as they appear not to represent this transformation. Based on your responses, I cannot find an exact match with this simplification.
1 answer