Factorization of polynomials
Answers to Questions
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
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.