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

    • Lösungsbeispiel

    Analyze x³ - 6x² + 11x - 6 = 0 (roots are 1, 2, 3).

    1. Sum of roots = -(-6)/1 = 6 (= 1+2+3)
    2. Sum of pairwise products = 11/1 = 11 (= 1×2+1×3+2×3)
    3. Product of roots = -(-6)/1 = 6 (= 1×2×3)