Partial fraction decomposition
Problem 4.395 · medium
Find the partial fraction decomposition of \( \displaystyle \frac{5 x - 4}{x^{3} + x^{2} + 2 x + 2} \).
- \[ \left(x + 1\right) \left(x^{2} + 2\right) \]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
- \[ \frac{5 x - 4}{x^{3} + x^{2} + 2 x + 2} = \frac{3 x + 2}{x^{2} + 2} - \frac{3}{x + 1} \]The decomposition; recombining it gives back the original fraction.✓ Proved
Answer \( \frac{3 x + 2}{x^{2} + 2} - \frac{3}{x + 1} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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: passqwen3.6:27b-mlx: pass — The solution correctly factors the denominator, sets up the appropriate partial fraction form, and provides the correct decomposition.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly factors the denominator, sets up the appropriate partial fraction form, and provides the correct decomposition.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the factors of the denominator and applies the standard partial fraction decomposition method. The final result is algebraically correct.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.