Partial fraction decomposition
Problem 4.694 · medium
Find the partial fraction decomposition of \( \displaystyle \frac{8 x + 1}{\left(x - 1\right) \left(x + 2\right)} \).
- \[ \left(x + 2\right) \left(x - 1\right) = \left(x - 1\right) \left(x + 2\right) \]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
- \[ \frac{8 x + 1}{\left(x - 1\right) \left(x + 2\right)} = \frac{5}{x + 2} + \frac{3}{x - 1} \]The decomposition; recombining it gives back the original fraction.✓ Proved
Answer \( \frac{5}{x + 2} + \frac{3}{x - 1} \)
✓ 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 is concise and correct. The setup sentence accurately describes the standard method for partial fraction decomposition with distinct linear factors, and the final equation is verified as correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution is concise and correct. The setup sentence accurately describes the standard method for partial fraction decomposition with distinct linear factors, and the final equation is verified as correct.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the form of the partial fraction decomposition and provides the correct coefficients. The steps are logically sound and the final answer is verified.
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 2026-10-08 with SymPy 1.14.0.