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 x^{4}+x^{2}+1 using only those identities.

Let us substitute x^2 = p

Then:

x^4 + x^2 + 1

= p^2 + p + 1

= p^2 + 2p + 1 - p

= (q-1)^2 + 2(q-1) + 1 - p [substituting q = p+1, so that p = q-1]

= (q-1)(q-1) + 2(q-1) + 1 - x^2 [resubstituting the value of p]

= q(q-1) - (q-1) + 2(q-1) + 1 - x^2

= q(q-1) + (q-1) + 1 - x^2

= (q-1)(q+1) + 1 - x^2

= (q^2 - 1) + 1 - x^2 [By Identity (i)]

= q^2 - x^2

= (q-x)(q+x) [By Identity (i)]

= (p-x+1)(p+x+1) [Resubstituting the value of q]

= (x^2-x+1)(x^2+x+1) [Resubstituting the value of p in the expression]

*Edited on ***March 22, 2007, 3:57 am**