12 May
2012
12 May
'12
9:23 a.m.
For those of us who've never heard of complex numbers x^4 + 4 = x^4 + 4x^2 + 4 - 4x^2 = (x^2 + 2 - 2x)(x^2 + 2 + 2x) Always complete the square! R. On Sat, 12 May 2012, Adam P. Goucher wrote:
E.g., for algebra, "x^4+4 doesn't factor, does it?"
x^4 + 4 = (x^2 + 2i)(x^2 - 2i) = (x + i + 1)(x - i - 1)(x - i + 1)(x + i - 1) = (x^2 + 2x + 2)(x^2 - 2x + 2)
So, yes, it does factor, and it only requires knowledge of the principal eighth roots of unity to do so.