Polynomial Roots Calculator Formula

Understand the math behind the polynomial roots calculator. Each variable explained with a worked example.

Use the Polynomial Roots Calculator

Formulas Used

Sum Roots

sum_roots = a != 0 ? -b / a : 0

Sum Products Pairs

sum_products_pairs = a != 0 ? c / a : 0

Product Roots

product_roots = a != 0 ? -d / a : 0

Discriminant

discriminant = a != 0 ? 18*a*b*c*d - 4*pow(b,3)*d + pow(b,2)*pow(c,2) - 4*a*pow(c,3) - 27*pow(a,2)*pow(d,2) : 0

Variables

VariableDescriptionDefault
aCoefficient a (x³)1
bCoefficient b (x²)-6
cCoefficient c (x)11
dCoefficient d (constant)-6

How It Works

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

    • Worked Example

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

    a = 1b = -6c = 11d = -6
    1. 01Sum of roots = -(-6)/1 = 6 (= 1+2+3)
    2. 02Sum of pairwise products = 11/1 = 11 (= 1×2+1×3+2×3)
    3. 03Product of roots = -(-6)/1 = 6 (= 1×2×3)

    Frequently Asked Questions

    What are Vieta's formulas?

    Vieta's formulas express relationships between the coefficients of a polynomial and sums and products of its roots, without needing to find the roots explicitly.

    Can this find the actual roots?

    This calculator shows the sum and product of roots via Vieta's formulas. For exact roots, factoring or numerical methods are typically needed for cubics and higher.

    What does the discriminant indicate?

    A positive discriminant means three distinct real roots. Zero means at least one repeated root. Negative means one real root and two complex conjugate roots.

    Learn More

    Guide

    How to Solve Quadratic Equations - Complete Guide

    Learn how to solve quadratic equations using factoring, the quadratic formula, and completing the square. Step-by-step methods with worked examples.

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