Question

If $\omega$ is a cube root of unity then $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ has a
factor

• A. $a+b\omega +c{\omega }^2$
• B. a-b-c
• C. $a-b{w}^2-cw$
• D. a+b-c

Answer 1

The correct option is A $a+b\omega +c{\omega }^2$

$△=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

$△=3abc-a^3-b^3-c^3$

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

Factors are $a+b+c,a+b\omega +c{\omega }^2,a+b{\omega }^2+c{\omega }^2$