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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.

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Polynomial Factorization -

Tag(s) : Symbolic Computation, Functions

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# Polynomial Factorization

## Factorization of polynomials

 Factorization Classic Classic (complex allowed) Full (with roots)

### 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.

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