Arbitrary Degree Polynomial Calculator

Equation Degree (1-10):

Coefficient x^2:

Coefficient x:

Constant Term:

Demo Examples

Introduction to Polynomial Equations

An nth-degree polynomial equation is the general form of algebraic equations containing the highest term xⁿ. According to the Fundamental Theorem of Algebra, an nth-degree equation has n roots (including complex roots).

Fundamental Theorem of Algebra

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

An nth-degree equation has n roots (including complex and repeated roots)

Solution Methods

Special Cases：

Linear equation: Direct solution
Quadratic equation: Quadratic formula
Cubic and higher: Numerical methods

Numerical Methods

xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)

Newton's Method Iteration

Introduction to Polynomial Equations

Solution Methods

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

f'(x) = naₙxⁿ⁻¹ + (n-1)aₙ₋₁xⁿ⁻² + ... + a₁

Numerical Methods

For cubic and higher equations, Newton's iterative method is used for numerical solution:

xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)

Convergence condition: |xₙ₊₁ - xₙ| < 10⁻¹⁰
Max iterations: 100
Multiple initial values: Using different starting points to find multiple roots

Demo Examples Description

2x - 3 = 0 (Linear equation) - Solution: x = 1.5
x² - 5x + 6 = 0 (Quadratic equation) - Solutions: x = 2, 3
x³ - 6x² + 11x - 6 = 0 (Cubic equation) - Solutions: x = 1, 2, 3
x⁴ - 10x² + 9 = 0 (Quartic equation) - Solutions: x = ±1, ±3