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 factoring all-one polynomials using the grouping method (Example)

The method of grouping terms can be used to factor all-one polynomials, i.e. polynomials of the form

when is composite. (When is prime, these polynomials are irreducible, so there is nothing to do in that case.)

Let us consider a few examples:

:

:

:

Combining this result with the factorization we have for the case , we obtain the following:

:

:

It might be worth pointing out that the polynomials produced by this factorization are not all irreducible. For instance,

However, to obtain this factorization, one needs to use some techique other than the grouping method. Likewise. the polynomial is also reducible.

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Cross-references: reducible, irreducible, prime, composite, polynomials, all-one polynomials, factor, terms

This is version 10 of factoring all-one polynomials using the grouping method, born on 2005-03-06, modified 2005-08-09.
Object id is 6851, canonical name is FactoringAllOnePolynomialsUsingTheGroupingMethod.
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Classification:
 AMS MSC: 13P05 (Commutative rings and algebras :: Computational aspects of commutative algebra :: Polynomials, factorization)