Polynomial Solver Step by Step8.7

Polynomial Solver Step by Step is a calculator, and, more importantly, a teaching tool.
It is designed to help students of both basic and advanced levels.

Basic Algebra functions:
1. Polynomial multiplication.
2. Polynomial long division with step by step explanation.
3. Algorithm for finding rational roots consisting of the following:
a. Finding all possible rational roots.
b. Eliminating the roots outside bounds.
c. Performing polynomial long division for each of the roots.
d. Adjusting bounds based on long division results if appropriate.

Iterative polynomial solving algorithms for advanced-level students:
1. Durand-Kerner method.
2. Aberth method.
3. Newton method with both real and complex starting points.
4. Global (also known as "damped") Newton method with both real and complex starting points.
5. Laguerre method with both real and complex starting points.
6. Bairstow method.
7. Muller method.
8. Eigenvalues of ‘companion matrix’ (demonstration of QR algorithm for eigenvalue calculation is available in our “Matrix Solver Step by Step" app).

Interactive graphs are available for:
1. Polynomial functions and their derivatives.
2. Aberth and Durand-Kerner method points on complex plane.
3. Newton, global Newton, Laguerre, and Muller algorithm steps.
4. Newton and global Newton method fractals / convergence images on complex plane
5. Laguerre method convergence images on complex plane
6 Bairstow method convergence images depending on initial approximation values.
7 Muller method convergence images depending on initial approximation values.
(Images 4-7 have adjustable settings for area, resolution and shading).

Links to online references are included.

Please write to us if you like to have new features / chapters added!

Developers welcome any feedback, please write to us at graphmath@aol.com.

Version History Link to our new statistical app, Discrete Probability (Calculator, Proofs, Graphs). 8.7 12/13/2025 Bug fix 8.6.1 09/15/2025 Link to our new app, 'Math Editor', which is a tool to create Math notes, posters and presentations. 8.6 04/25/2025 Fixed an issue with e-mail not sending from "About / Contact us" section. 8.5 04/14/2025 Minor changes 8.4 03/09/2025 Added a link to our new app, "Construct Polynomial: Interpolate and Approximate" 8.3 02/03/2025 Bug fix 8.2.1 01/23/2025 1. Explanation of Durand-Kerner method.
2. UI improvements, performance improvements and bug fixes. 8.2 01/22/2025 UI improvements and bug fixes. 8.1 01/16/2025 Muller method for iterative polynomial solution, including convergence images depending on initial approximation values. 8.0 12/30/2024 Bairstow method convergence image improvements. 7.1 12/28/2024 Bug fix. 7.0.2 12/26/2024 Bug fixes. 7.0.1 12/24/2024 1. Bairstow method, an iterative method that finds a pair of polynomial roots, given starting values u0 and v0.
2. Bairstow method convergence images.
3. Improvements in calculation efficiency and image presentation. 7.0 12/23/2024 Bug fixes. 6.1 12/19/2024 Crash prevention. 6.0.1 12/15/2024 1. Approximation by Laguerre's method.
2. Complex plane image by Laguerre's method.
3. Performance and UI improvements, bug fixes. 6.0 12/13/2024 Bug fix 5.2.1 12/11/2024 Calculation performance and precision improvements, bug fixes. 5.2 12/10/2024 1. To improve accuracy, polynomials are deflated by repeated roots before applying Aberth algorithm.
2. Bug fixes and UI improvements. 5.1 12/09/2024 Bug fix. 5.0.1 12/07/2024 1. Polynomial multiplication.
2. Polynomial long division.
3. UI improvements and bug fixes. 5.0 12/06/2024 1. Finding repeated roots.
2. UI and numeric precision improvements. 4.1 12/02/2024 1. Images of Newton and global Newton fractals on complex plane with adjustable settings for area, resolution and shading.
2. UI improvements and bug fixes. 4.0 11/27/2024 1. Global (also known as "damped") Newton method for improved convergence rate and faster convergence.
2. UI improvements and bug fixes. 3.0 11/26/2024