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Partial fraction decomposition

Problem 4.540 · medium

Find the partial fraction decomposition of \( \displaystyle \frac{26 - 7 x}{x^{2} - 7 x + 12} \).

\[ \left(x - 3\right) \left(x - 4\right) = \left(x - 4\right) \left(x - 3\right) \]
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
\[ \frac{26 - 7 x}{x^{2} - 7 x + 12} = - \frac{5}{x - 3} - \frac{2}{x - 4} \]
The decomposition; recombining it gives back the original fraction.✓ Proved

Answer \( - \frac{5}{x - 3} - \frac{2}{x - 4} \)

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 identifies the factors of the denominator and provides the correct partial fraction decomposition. The intermediate steps are omitted but the final result is verified as correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the factors of the denominator and provides the correct partial fraction decomposition. The intermediate steps are omitted but the final result is verified as correct.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to show the derivation of the coefficients (solving for A and B), jumping directly to the final answer. While the final answer is correct, the 'solution' does not demonstrate the method requested by the problem statement.

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 2026-10-06 with SymPy 1.14.0.