Polynomial Roots Solver (Up to Degree 4)

Polynomial Roots Calculator 2025: Complete Guide

Solve any polynomial up to degree 4: x⁴ + x² - 3x + 2 = 0. Get real & complex roots, factoring, graph, and full step-by-step.

How to Solve Polynomials

  1. Identify degree and coefficients
  2. Use factoring, quadratic formula, or numerical methods
  3. Find all roots (real + complex)
  4. Verify with graph

Example: x² - 3x + 2 = 0

(x - 1)(x - 2) = 0
x = 1 or x = 2

Roots Types

  • Real Roots: Cross x-axis
  • Complex Roots: Come in conjugate pairs
  • Repeated Roots: Multiplicity > 1

Factoring Methods

  • Quadratic: ax² + bx + c = 0 to x = [-b ± √(b²-4ac)] / 2a
  • Cubic: Try rational roots
  • Quartic: Factor into quadratics

Graph Behavior

  • Degree 2: Parabola
  • Degree 3: S-shape
  • Degree 4: W or M shape

Common Mistakes

  • Forgetting complex roots
  • Wrong sign in quadratic formula
  • Missing repeated roots

Real-World Uses

  • Physics: Motion equations
  • Engineering: Signal processing
  • Economics: Cost functions
  • Computer Graphics: Curves

Advanced Tools

FAQs

Can it find complex roots? Yes — always in conjugate pairs.

Degree higher than 4? No closed-form solution (Abel-Ruffini).

Can I graph it? Yes — interactive plot with roots marked.

Conclusion

Polynomials model everything from physics to finance. Our 2025 Polynomial Roots Calculator handles up to degree 4 with full steps, factoring, and graph. Solve confidently!

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