Find real and complex roots in seconds
Enter coefficients from highest degree to constant term. Use commas or spaces. Example: 1, -6, 11, -6 for x³ - 6x² + 11x - 6 = 0.
What is a polynomial root calculator?
A polynomial root calculator helps you solve equations of the form:
anxn + an-1xn-1 + ... + a1x + a0 = 0
The numbers that make this equation true are called roots (or zeros). Depending on the polynomial, roots may be real or complex. This tool computes both.
How to use this calculator
Step 1: Enter coefficients
Type coefficients in descending power order. For example:
- 2, -8 → 2x - 8 = 0
- 1, 0, -9 → x² - 9 = 0
- 1, -6, 11, -6 → x³ - 6x² + 11x - 6 = 0
Step 2: Choose precision
Precision controls how many decimal places are shown in your answers.
Step 3: Click “Calculate Roots”
The calculator returns each root and a small residual check value |P(r)| to indicate numerical accuracy.
How results are computed
Linear equations (degree 1)
Calculated directly with x = -b/a.
Quadratic equations (degree 2)
Calculated using the quadratic formula, including complex values when the discriminant is negative.
Higher-degree equations (degree 3+)
This page uses an iterative numerical method (Durand–Kerner) to approximate all roots, including complex roots. It is fast and practical for most everyday polynomials.
Real roots vs complex roots
Not every polynomial has only real solutions. For example:
- x² - 1 = 0 has real roots: 1 and -1
- x² + 1 = 0 has complex roots: i and -i
Complex roots are shown in the form a ± bi.
Common input mistakes
- Wrong coefficient order (must be highest power to constant)
- Missing middle terms without placing 0 (e.g., x³ - 5 should be 1, 0, 0, -5)
- Using variables or symbols instead of numeric coefficients
Where polynomial roots are used
- Control systems and stability analysis
- Signal processing and filter design
- Physics and engineering models
- Computer graphics and curve intersections
- Optimization and numerical analysis
Quick takeaway
If you need roots quickly, this calculator gives a practical answer with real and complex solutions, clear formatting, and a built-in accuracy check. For repeated roots or very high-degree sensitive polynomials, slight numerical variation is normal.