Search for a tool
Polynomial Factorization

Tool for Factorization of a polynomial. Factorizing consists in expressing a polynomial as a product, so it can be it's canonical form.

Results

Polynomial Factorization -

Tag(s) : Symbolic Computation, Functions

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Thanks to your feedback and relevant comments, dCode has developed the best 'Polynomial Factorization' tool, so feel free to write! Thank you!

Polynomial Factorization

Factorization of polynomials





Answers to Questions (FAQ)

How to factorize a polynomial-like expression?

Factorizing a mathematical polynomial expression of degree $ n $ means to express it as a product of polynomial factors.

Among the polynomial factorization's methods, the simplest is to recognize a remarkable identity. Remarkables identities also apply with polynomials

Example: $ a^2+2ab+b^2 $ is a 2nd order polynomial that factorizes as $ (a+b)^2 $

Example: $ x^2+2x-a^2+1 $ is factorized $ (x-a+1)(x+a+1) $

Another method is to try variable values like $ x = 0, 1, -1, 2, -2 $, which are sometimes the polynomials roots and allow you to find solutions quickly.

Example: $ x^2-4 $ has the root $ -2 $ and $ 2 $ and thus can be factorized $ (x-2)(x+2) $

Do not confuse with the canonical form of a polynomial

How to factorize a 2nd degree polynomial?

Method 1: Find remarkable identities.

Example: $ x^2+2x+1 $ is factored $ (x+1)^2 $

Method 2: Calculate the roots of the polynomial, a second degree polynomial $ P $ having 2 roots $ a $ and $ b $ is factored $ P = (x-a) (x-b) $

Example: $ p = x^2-4x-5 $ has 2 roots: $ x = 5 $ and $ x = -1 $, it cam be factorized as $ p = (x-5)(x+1) $

How to factorize a 3rd degree polynomial?

Method 1: by knowing a root $ a $ of the polynomial $ p $ (possibly an obvious root), then the polynomial can be factored by $ (x−a) $, that is $ p = (x−a) \cdot q(x) $ avec $ q(x) $ a polynomial of degree 2 (factorization method above).

Method 2: knowing its 3 roots $ a, b, c $ then $ p = (x-a)(x-b)(x-c) $

How to factorize a Nth degree polynomial?

Method 1: by finding/knowing a root $ a $ of the polynomial $ p $, then the polynomial can be factored by $ (x−a) $, that is $ p = (x−a) \cdot q(x) $ with $ q(x) $ a polynomial of degree $ n - 1 $. Reapply this method on the polynomial $ q $ iteratively.

Method 2: knowing all the roots $ a_1, a_2, a_3 \cdots \a_n $ then $ p = (x-a_1)(x-a_2)\cdots(x-a_n) $ (some roots can be identical)

Method 3: use the dCode solver at the top of this page.

How to factorize a 4th or 5th or 6th degree polynomial?

Apply the method to factor a polynomial of degree $ n $ (above) or use the dCode solver at the top of this page.

What is a remarkable identity?

A remarkable identity is an equality demonstrated between two mathematical terms, which is common enough to be detectable and usable without further demonstration. The best known are those used in factoring polynomials of degree 2:

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

$$ (a-b)^2 = a^2 - 2ab + b^2 $$

$$ (a+b)(a-b)=a^2 - b^2 $$

What is an irreducible polynomial?

Irreducible polynomials are polynomials which cannot be decomposed into a product of two non-constant polynomials.

1st Degree polynomials are always irreducible.

Source code

dCode retains ownership of the online 'Polynomial Factorization' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Polynomial Factorization' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Polynomial Factorization' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Polynomial Factorization' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Thanks to your feedback and relevant comments, dCode has developed the best 'Polynomial Factorization' tool, so feel free to write! Thank you!


Source : https://www.dcode.fr/polynomial-factorization
© 2021 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.
Feedback