Polynomwurzelrechner — Formel
## Polynomial Roots via Vieta's Formulas
### For a Cubic ax³ + bx² + cx + d = 0
Vieta's formulas relate the roots (r₁, r₂, r₃) to the coefficients:
- **r₁ + r₂ + r₃ = -b/a** (sum of roots)
- **r₁r₂ + r₁r₃ + r₂r₃ = c/a** (sum of pairwise products)
- **r₁r₂r₃ = -d/a** (product of roots)
### Discriminant
The discriminant tells you about the nature of the roots:
- Δ > 0: three distinct real roots
- Δ = 0: a repeated root
- Δ < 0: one real root and two complex conjugate roots
### For a Cubic ax³ + bx² + cx + d = 0
Vieta's formulas relate the roots (r₁, r₂, r₃) to the coefficients:
- **r₁ + r₂ + r₃ = -b/a** (sum of roots)
- **r₁r₂ + r₁r₃ + r₂r₃ = c/a** (sum of pairwise products)
- **r₁r₂r₃ = -d/a** (product of roots)
### Discriminant
The discriminant tells you about the nature of the roots:
- Δ > 0: three distinct real roots
- Δ = 0: a repeated root
- Δ < 0: one real root and two complex conjugate roots
Lösungsbeispiel
Analyze x³ - 6x² + 11x - 6 = 0 (roots are 1, 2, 3).
- Sum of roots = -(-6)/1 = 6 (= 1+2+3)
- Sum of pairwise products = 11/1 = 11 (= 1×2+1×3+2×3)
- Product of roots = -(-6)/1 = 6 (= 1×2×3)