When math students are taught factoring, usually the first thing they learn are special factoring identities such as:

1: a²-b²= (a-b)(a+b)
2: a³-b³ = (a-b)(a²+ab+b²)
3: a³+b³ = (a+b)(a²-ab+b²)

Find a factorization of the expression x⁴+x²+1 using only those identities.

x^4 + x² + 1
x^4 - x² + 2x² + x - x + 1  [add zero as -x² + x² + x - x]
(x² + x)(x² - x) + x² + x + x² - x + 1  [rule 1]
(x² + x)(x² - x) + (x² + x) + (x² - x + 1) [group terms]
(x² + x)(x² - x + 1) + (x² - x + 1) [monomial factor] 
(x² + x +1)(x² - x + 1) [monomial factor]

I assume this is ok as the factoring either involves rule 1 or does not involve an identity.

Posted by Jer on 2007-03-21 11:07:37