Try multiple initial guesses to find different roots. Newton finds the nearest root to x₀.

Initial guess with f'(x₀) = 0 → division by zero. Cycling between two values (rare). Diverging (poor guess).

When to prefer bisection?

When you need guaranteed convergence (slower). Newton is faster when it works.

Generalized Newton uses Jacobian inverse. Used in optimization (gradient descent variants).

Newton's Method Root Finder

xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ). Iteratively find roots of polynomial.

Result

Root estimate

1.32471817

f(root) ≈ 0.00000092; 3 iterations.

f(x)1x³ + 0x² + -1x + -1
Initial x₀1.5
Final x1.32471817
f(final x)0.00000092
Iterations3
ConvergedYes
Iter 0: x=1.500000, f(x)=0.875000
Iter 1: x=1.347826, f(x)=0.100682
Iter 2: x=1.325200, f(x)=0.002058
Iter 3: x=1.324718, f(x)=0.000001

Step-by-step

Newton iteration: xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ).
Converged in 3 iterations to x = 1.324718.

How to use this calculator

Enter polynomial coefficients + initial guess.

About this calculator

Newton's method for finding roots: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Quadratic convergence near a root (each iteration roughly doubles correct digits). Requires good initial guess + non-zero derivative. Can fail: jumping between roots, oscillating, dividing by zero (at critical points). Used everywhere in numerical analysis, optimization (find zero of gradient), implicit equation solving. Source: Wolfram MathWorld - Newton's Method.