Formulae and Techniques Used
- Content extraction: factor out gcd of coefficients to get a primitive polynomial.
- Rational Root Theorem: possible roots are ± p/q where p|a0, q|an.
- Factor Theorem: if r is a root, then (x − r) divides the polynomial.
- Exact division by (q x − p): integer synthetic division checks remainder zero.
- Special identities: a2 − b2 = (a−b)(a+b); a3 ± b3 = (a±b)(a2 ∓ ab + b2).
- Quadratic check: discriminant b2−4ac detects reducible quadratics over the rationals.
How to Use This Calculator
- Enter your polynomial using one variable (e.g., x). Fractions allowed.
- Optionally set the variable letter; otherwise, it auto-detects.
- Click Factorize to compute the complete factorization over the rationals.
- Review normalized form, extracted content, and factors with multiplicities.
- Export the results: download CSV, or print/save as PDF.
Supported Degrees and Factor Forms
Designed for single-variable polynomials up to moderate degree. Extracts all rational linear factors and detects common identities including difference of squares and sum/difference of cubes when applicable.
- Coefficients: integers or rational numbers (fractions).
- Roots: rational candidates via ±p/q where p|constant, q|leading.
- Irreducible remainder shown as a parenthesized polynomial factor.
Limitations and Notes
- Only one variable is supported per computation.
- General irreducible cubics/quartics without rational roots remain unfactored.
- If denominators appear, the parser clears them to use integer arithmetic.
- Repeated factors are automatically consolidated with multiplicities.
Example Data
| # | Polynomial | Variable | Expected Factorization (over ℚ) | Notes |
|---|---|---|---|---|
| 1 | x^2 - 5x + 6 | x | (x - 2)(x - 3) | Two rational roots |
| 2 | 2x^3 + x^2 - 8x - 4 | x | (2x + 1)(x - 2)(x + 2) | Three rational roots |
| 3 | x^4 - 1 | x | (x - 1)(x + 1)(x^2 + 1) | Difference of squares twice |
| 4 | 8x^3 - 27 | x | (2x - 3)(4x^2 + 6x + 9) | Difference of cubes |
| 5 | 6x^2 + 5x - 6 | x | (3x - 2)(2x + 3) | Quadratic with rational factorization |
Factorization Patterns Reference
| Pattern | Algebraic Form | Factors (over ℚ) | Example |
|---|---|---|---|
| Difference of squares | a2 − b2 | (a − b)(a + b) | x4 − 1 → (x2 − 1)(x2 + 1) |
| Sum of cubes | a3 + b3 | (a + b)(a2 − ab + b2) | x3 + 8 → (x + 2)(x2 − 2x + 4) |
| Difference of cubes | a3 − b3 | (a − b)(a2 + ab + b2) | 8x3 − 27 → (2x − 3)(4x2 + 6x + 9) |
| Perfect square trinomial | a2 ± 2ab + b2 | (a ± b)2 | x2 + 6x + 9 → (x + 3)2 |
| Quadratic reducibility | ax2 + bx + c | Rational roots exist if Δ = b2 − 4ac is a rational square | 6x2 + 5x − 6 → (3x − 2)(2x + 3) |
| Grouping (four terms) | ax3 + bx2 + cx + d | Group pairs to expose a common binomial factor | 2x3 + x2 − 8x − 4 → (2x + 1)(x − 2)(x + 2) |
Rational Root Candidates — Counts
For a primitive polynomial with leading coefficient |an| and constant term |a0|, candidates are ±p/q where p | |a0| and q | |an|. The maximum number of candidates is 2·τ(|a0|)·τ(|an|), where τ(n) counts the positive divisors of n.
| |a0| | |an| | τ(|a0|) | τ(|an|) | Max Candidates 2·τ(|a0|)·τ(|an|) | Notes |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 2 | Only ±1 |
| 6 | 1 | 4 | 1 | 8 | ±1, ±2, ±3, ±6 |
| 6 | 2 | 4 | 2 | 16 | Denominators 1 or 2 |
| 12 | 3 | 6 | 2 | 24 | Many ratios reduce |
| 30 | 6 | 8 | 4 | 64 | Large candidate set |
FAQs
It means factors use integer coefficients and rational numbers only. Complex or irrational factors are left inside irreducible polynomial blocks.
Yes. The parser multiplies by the least common multiple of denominators to clear fractions before factorization.
Repeated factors are automatically consolidated and displayed with exponents indicating multiplicity.
No. If no rational factors exist, the remaining polynomial is shown as an irreducible factor over the rationals.
Difference of squares, sum and difference of cubes, and simple grouping for some four-term polynomials.
No. Terms are parsed and combined by exponent. The calculator outputs a normalized polynomial and sorts terms by degree automatically.
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