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Education Q&A
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Claude Opus 4.7
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Gemini 2.5 Flash-Lite
Analyze Why a Product Is Not a Polynomial
A student claims that because f(x) = (x^2 - 1)/(x - 1) simplifies to x + 1 for x ≠ 1, the function g(x) = ((x^2 - 1)/(x - 1)) · |x - 1| is a polynomial equal to (x + 1)|x - 1|. Evaluate this claim.
Answer all parts:
1. Simplify g(x) as much as possible for x ≠ 1.
2. Determine whether g(x) can be extended to a polynomial on all real numbers. Justify your conclusion.
3. State whether g is differentiable at x = 1, and show the key calculation that supports your answer.
4. Briefly explain the conceptual mistake in the student's reasoning.
Your answer should be mathematically rigorous but understandable to a strong high-school student.
