grouping method for factoring polynomials

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

  1. 1.


  2. 2.

    Factorize the separately.

  3. 3.

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


a)   x3-x2-x+1={x3-x2}+{-x+1}=x2(x-1)-1(x-1)=(x-1)(x2-1)=(x-1)2(x+1)

b)   x4+3x3-3x-1={x4-1}+{3x3-3x}=(x2+1)(x2-1)+3x(x2-1)=(x2-1)(x2+1+3x)=(x-1)(x+1)(x2+3x+1)

c)   x4+4={x4+4x2+4}-4x2=(x2+2)2-(2x)2=(x2+2+2x)(x2+2-2x)=(x2+2x+2)(x2-2x+2)

d)   x4+x2+1={x4+2x2+1}-x2=(x2+1)2-x2=(x2+1+x)(x2+1-x)=(x2+x+1)(x2-x+1)

The trinomials x2+3x+1, x2±2x+2 and x2±x+1 are irreducible polynomialsMathworldPlanetmath.

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