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.