Analyze Why a Product Is Not a Polynomial

The answer is entirely correct across all four parts. The simplification, the piecewise definition, the one-sided derivatives (2 and -2), and the final conclusion are all accurate.

The reasoning is exceptionally strong. The use of the identity theorem for polynomials in part 2 is a highly rigorous and insightful argument. The answer also provides a second, independent proof based on non-differentiability, demonstrating a deep understanding of the concepts.

The answer is fully complete, addressing every component of the four-part prompt in detail. It provides the simplified form, the piecewise definition, the full derivative calculation, and a thorough conceptual explanation.

The answer is exceptionally clear and well-organized. The use of bolded headers and bullet points makes the structure easy to follow. The explanation of the complex polynomial identity argument is particularly lucid.

The answer perfectly follows all instructions. It addresses all four parts, provides justifications, and is written in a style appropriate for the target audience.

All mathematical claims are correct: the simplification, the piecewise form, the contradiction proof using polynomial identity on intervals, the one-sided derivative calculations (2 and -2), and the conclusion about non-differentiability. No errors detected.

Exceptional reasoning. Uses the polynomial identity principle (two polynomials agreeing on an infinite set are identical) to give a clean contradiction proof. Also provides a second independent argument via non-differentiability. The logic is tight and multi-layered.

All four parts are addressed fully and in depth. The extension discussion explicitly handles the undefined point at x=1 and the filled-in value. Both one-sided derivatives are computed with full algebraic detail.

Well-organized with bold headers, clear case splits, and explicit limit calculations. The language is precise yet accessible. The two-proof structure for non-polynomiality is clearly signposted.

Follows all four parts of the prompt in order, addresses the expected content (simplification, extension, differentiability, conceptual mistake), and maintains the appropriate level for a strong high-school student.

All core claims are correct: the simplification for x ≠ 1, the piecewise forms, the nonexistence of a polynomial extension, and the one-sided derivative calculation at x = 1. It also correctly distinguishes the original undefined point from the continuous extension. A tiny notation inconsistency appears in one displayed piecewise line using x ≤ 1 after earlier deriving x < 1, but the intended extension is still mathematically clear.

Reasoning is rigorous and well structured. It gives a contradiction argument using the identity principle for polynomials on intervals and also notes non-differentiability as an independent obstruction. The conceptual distinction between equality on a restricted domain and equality as functions is especially strong.

It fully answers every requested part, including simplification, extension analysis, differentiability, and the conceptual mistake. It also adds helpful supporting remarks about continuity and gives two different ways to see non-polynomiality.

The explanation is clear, logically ordered, and pitched appropriately for a strong high-school student while remaining rigorous. The case split and derivative computation are especially readable.

It follows the prompt very closely, answers all numbered parts explicitly, and maintains a rigorous but accessible essay style. It also directly evaluates the student's claim rather than merely restating computations.

The answer is entirely correct. All calculations, including the one-sided derivatives (2 and -2), are accurate, and the conclusions for each part of the prompt are correct.

The reasoning is correct and logical. However, the argument in part 2, while correct, is less mathematically rigorous than Answer A's. It relies on the intuitive idea of a 'sharp corner' rather than a more formal proof like the polynomial identity theorem.

The answer is fully complete. It systematically addresses all four questions posed in the prompt, providing detailed calculations and explanations for each.

The answer is very clear and well-written. The steps for each calculation are laid out logically, making it easy to follow. The use of LaTeX for mathematical notation enhances readability.

The answer perfectly adheres to the prompt's instructions. It answers all four parts, provides the required calculations and justifications, and maintains a tone suitable for a strong high-school student.

All core mathematical claims are correct: simplification, piecewise form, one-sided derivatives, and conclusion. However, the non-polynomiality argument in part 2 is informal (relies on 'sharp corner' intuition rather than a rigorous algebraic proof), which slightly weakens the correctness of the justification even if the conclusion is right.

Reasoning is adequate but not deep. The non-polynomiality argument rests on the observation that absolute value creates a piecewise definition and a 'sharp corner,' which is intuitive but not a rigorous proof. No use of the polynomial identity principle or other formal argument.

All four parts are addressed. The extension discussion covers the piecewise form and mentions the undefined point. One-sided derivatives are computed correctly. Slightly less depth in the conceptual explanation but overall complete.

Clear and readable with good use of LaTeX formatting. The structure follows the four parts cleanly. Slightly less polished in transitions between ideas, but overall clear.

Follows all four parts of the prompt in order and addresses each required element. Slightly less explicit about the 'removable hole' distinction but otherwise well-aligned with the prompt requirements.

The main conclusions are correct: g(x) = (x + 1)|x - 1| for x ≠ 1, the piecewise forms x^2 - 1 and 1 - x^2 are correct, and the derivative test at x = 1 is computed correctly. The weaker point is the argument about polynomial extension, which is stated correctly but not proved as rigorously as it could be.

Reasoning is generally sound and easy to follow, but the argument against polynomial extension leans on the presence of absolute value and a corner rather than fully proving impossibility through polynomial uniqueness on intervals. This is acceptable but less robust for a mathematically rigorous benchmark response.

It addresses all four requested parts and includes the key derivative calculation. The only missing element relative to a stronger benchmark answer is a more complete proof that no single polynomial can match both interval formulas.

The answer is clear and readable, with a straightforward step-by-step structure. Some phrasing is a bit repetitive, and the conceptual distinction between the original function and an extension could be made more sharply.

It follows the requested structure and answers each numbered part in an understandable way. It is slightly less aligned with the prompt's emphasis on rigorous justification for the polynomial-extension question.