Question

Factorize:

$a^2-ab+b^2$

Answer 1

Now,

$a^2-ab+b^2$

$=a^2+(\omega +{\omega }^2)ab+b^2$ [ Since $1+\omega +{\omega }^2=0$ Where $\omega$ being cube root of unity]

$=a^2+(\omega +{\omega }^2)ab+b^2{\omega }^3$ [ Since ${\omega }^3=1$]

$=a(a+b\omega )+b{\omega }^2(a+b\omega )$

$=(a+b\omega )(a+b{\omega }^2)$.