When math students are taught factoring, usually the first thing they learn are special factoring identities such as:
1: a²b²= (ab)(a+b)
2: a³b³ = (ab)(a²+ab+b²)
3: a³+b³ = (a+b)(a²ab+b²)
Find a factorization of the expression x^{4}+x^{2}+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 20070321 11:07:37 