grouping method for factoring polynomials

Factoring a given polynomial may in certain special cases by using the following grouping method:

1. .

2. Factorize the separately.

3. The whole polynomial may then possibly be written in form of a product.

Examples

a)   $x^3-x^2-x+1=\{x^3-x^2\}+\{-x+1\}=x^2(x-1)-1(x-1)=(x-1)(x^2-1)={(x-1)}^2(x+1)$

b)   $x^4+3x^3-3x-1=\{x^4-1\}+\{3x^3-3x\}=(x^2+1)(x^2-1)+3x(x^2-1)=(x^2-1)(x^2+1+3x)=(x-1)(x+1)(x^2+3x+1)$

c)   $x^4+4=\{x^4+4x^2+4\}-4x^2={(x^2+2)}^2-{(2x)}^2=(x^2+2+2x)(x^2+2-2x)=(x^2+2x+2)(x^2-2x+2)$

d)   $x^4+x^2+1=\{x^4+2x^2+1\}-x^2={(x^2+1)}^2-x^2=(x^2+1+x)(x^2+1-x)=(x^2+x+1)(x^2-x+1)$

The trinomials $x^2+3x+1$, $x^2\pm 2x+2$ and $x^2\pm x+1$ are irreducible polynomials.

Title	grouping method for factoring polynomials
Canonical name	GroupingMethodForFactoringPolynomials
Date of creation	2013-03-22 15:06:49
Last modified on	2013-03-22 15:06:49
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	12
Author	pahio (2872)
Entry type	Algorithm
Classification	msc 13P05
Related topic	DifferenceOfSquares
Related topic	ExampleOfGcd
Related topic	ZeroRuleOfProduct
Defines	grouping method