Partial fraction decomposition
Problem 4.843 · medium
Find the partial fraction decomposition of \( \displaystyle \frac{- 7 x^{2} - 53 x - 99}{x^{3} + 11 x^{2} + 40 x + 48} \).
- \[ \left(x + 3\right) \left(x + 4\right)^{2} \]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
- \[ \frac{- 7 x^{2} - 53 x - 99}{x^{3} + 11 x^{2} + 40 x + 48} = - \frac{4}{x + 4} - \frac{1}{\left(x + 4\right)^{2}} - \frac{3}{x + 3} \]The decomposition; recombining it gives back the original fraction.✓ Proved
Answer \( - \frac{4}{x + 4} - \frac{1}{\left(x + 4\right)^{2}} - \frac{3}{x + 3} \)
✓ 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 partial fraction decomposition with the appropriate terms for the repeated linear factor, and provides the correct coefficients.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly factors the denominator, sets up the partial fraction decomposition with the appropriate terms for the repeated linear factor, and provides the correct coefficients.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the factorization of the denominator and the resulting partial fraction decomposition. The stated answer matches the decomposition shown in the final equation.
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 2026-10-10 with SymPy 1.14.0.