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Partial fraction decomposition

Problem 4.390 · medium

Find the partial fraction decomposition of \( \displaystyle \frac{x^{2} - 10 x + 8}{\left(x + 2\right) \left(x^{2} + 4\right)} \).

\[ \left(x + 2\right) \left(x^{2} + 4\right) \]
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
\[ \frac{x^{2} - 10 x + 8}{\left(x + 2\right) \left(x^{2} + 4\right)} = \frac{- 3 x - 4}{x^{2} + 4} + \frac{4}{x + 2} \]
The decomposition; recombining it gives back the original fraction.✓ Proved

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

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 identifies the form of the partial fraction decomposition and verifies the result algebraically. The stated answer matches the derived decomposition.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the form of the partial fraction decomposition and verifies the result algebraically. The stated answer matches the derived decomposition.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution skips the actual derivation steps (setting up the system of equations and solving for coefficients), presenting the final answer as a given fact rather than a result of the stated method.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.