.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/abstract_algebra/polynomial_ring/plot_01_division_and_gcd.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_abstract_algebra_polynomial_ring_plot_01_division_and_gcd.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_abstract_algebra_polynomial_ring_plot_01_division_and_gcd.py:


Polynomial long division and the Euclidean algorithm
============================================================

Divides two polynomials with remainder, and finds their greatest
common divisor via the polynomial Euclidean algorithm -- exactly the
integer Euclidean algorithm with polynomial division in place of
integer division.

.. GENERATED FROM PYTHON SOURCE LINES 12-14

.. code-block:: Python

    from mathematicskit.abstract_algebra import Polynomial, poly_divmod, poly_gcd, poly_mul








.. GENERATED FROM PYTHON SOURCE LINES 15-17

Divide x^3 - 1 by x - 1
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 17-24

.. code-block:: Python


    p = Polynomial([-1, 0, 0, 1])  # x^3 - 1
    q = Polynomial([-1, 1])  # x - 1
    quotient, remainder = poly_divmod(p, q)
    print(f"(x^3 - 1) / (x - 1) = quotient {quotient.coeffs}, remainder {remainder.coeffs}")
    print(f"check: quotient * (x-1) + remainder = {(poly_mul(quotient, q) + remainder).coeffs}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    (x^3 - 1) / (x - 1) = quotient [Fraction(1, 1), Fraction(1, 1), Fraction(1, 1)], remainder [Fraction(0, 1)]
    check: quotient * (x-1) + remainder = [Fraction(-1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)]




.. GENERATED FROM PYTHON SOURCE LINES 25-27

GCD of x^4 - 1 and x^6 - 1
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-32

.. code-block:: Python


    a = Polynomial([-1, 0, 0, 0, 1])  # x^4 - 1
    b = Polynomial([-1, 0, 0, 0, 0, 0, 1])  # x^6 - 1
    g = poly_gcd(a, b)
    print(f"\ngcd(x^4 - 1, x^6 - 1) has degree {g.degree}, coefficients {g.coeffs}")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    gcd(x^4 - 1, x^6 - 1) has degree 2, coefficients [Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1)]





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_abstract_algebra_polynomial_ring_plot_01_division_and_gcd.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_division_and_gcd.ipynb <plot_01_division_and_gcd.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_division_and_gcd.py <plot_01_division_and_gcd.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_division_and_gcd.zip <plot_01_division_and_gcd.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_