PlanetMath: grouping method for factoring polynomials

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grouping method for factoring polynomials

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

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.

"grouping method for factoring polynomials" is owned by pahio.

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Attachments:: factoring all-one polynomials using the grouping method (Example) by rspuzio

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Cross-references: irreducible polynomials, trinomials, product, polynomial
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This is version 9 of grouping method for factoring polynomials, born on 2005-03-06, modified 2008-02-28.
Object id is 6850, canonical name is GroupingMethodForFactorizingPolynomials.
Accessed 7559 times total.

Classification:

13P05 (Commutative rings and algebras :: Computational aspects of commutative algebra :: Polynomials, factorization)

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