Complex Polynomial Solver
Polynomial Degree:
2
Roots:
Solution Steps:
Understanding Complex Polynomials
What are Complex Polynomials?
Complex polynomials are expressions with complex coefficients:
P(z) = aₙzⁿ + aₙ₋₁zⁿ⁻¹ + ... + a₁z + a₀
where aᵢ = x + yi (complex numbers)
Fundamental Theorem of Algebra
Root Existence
Every polynomial has exactly n complex roots
Conjugate Pairs
Real coefficients yield conjugate complex roots
Multiplicity
Roots can have multiple occurrences
Factorization
Complete factorization always possible
Numerical Methods
Newton's Method
- Quadratic convergence
- Derivative required
- Initial guess sensitive
- Single root at a time
Laguerre's Method
- Cubic convergence
- Multiple derivatives
- Better convergence
- Complex arithmetic
Durand-Kerner
- Simultaneous roots
- No derivatives needed
- Global convergence
- Parallel computation
Special Cases
| Polynomial Type | Root Properties | Example |
|---|---|---|
| Linear | One root | z + 1 = 0 |
| Quadratic | Two roots | z² + 2z + 1 = 0 |
| Pure Complex | Symmetric roots | z² + i = 0 |
Real-World Applications
Control Systems
System stability analysis and pole placement
Signal Processing
Filter design and frequency analysis
Quantum Mechanics
Wave function solutions and energy levels