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Partial fraction decomposition

Problem 4.984 · medium

Find the partial fraction decomposition of \( \displaystyle \frac{2 x^{2} - 6 x - 10}{x^{3} - 2 x^{2} + 3 x - 6} \).

\[ \left(x - 2\right) \left(x^{2} + 3\right) \]
Factor the denominator.✓ Proved
Set up one term per factor (a repeated factor gets a term for each power; an irreducible quadratic gets Bx + C on top), clear denominators, and match coefficients.
Reviewed
\[ \frac{2 x^{2} - 6 x - 10}{x^{3} - 2 x^{2} + 3 x - 6} = \frac{4 x + 2}{x^{2} + 3} - \frac{2}{x - 2} \]
The decomposition; recombining it gives back the original fraction.✓ Proved

Answer \( \frac{4 x + 2}{x^{2} + 3} - \frac{2}{x - 2} \)

Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	SymPy's own apart() gives the same decomposition

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly factors the denominator, sets up the appropriate partial fraction form, and provides the verified decomposition.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly factors the denominator, sets up the appropriate partial fraction form, and provides the verified decomposition.
gpt-oss:20b: pass 2026-10-12
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution provides the final answer but skips the actual derivation steps (setting up the unknown coefficients, solving the system of equations). It presents the result as a given rather than demonstrating the method requested by the problem type.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/partial_fraction_decomp, checked 2026-10-12 with SymPy 1.14.0.