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  4. If f(x)=a+b x+c x2 and α, β, γ are roots of the equation x3=1, then |a&b&c b&c&a c&a&b| is equal to

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Q. If $f(x)=a+b x+c x^{2}$ and $\alpha, \beta, \gamma$ are roots of the equation $x^3=1$, then $ \begin{vmatrix}a&b&c\\ b&c&a\\ c&a&b\end{vmatrix}$ is equal to

Determinants

Solution:

$ \begin{vmatrix}a&b&c\\ b&c&a\\ c&a&b\end{vmatrix} =-(a^{3}+b^3+c^3-3abc)$
$=-(a+b+c)\left(a+b \omega^{2}+c \omega\right)\left(a+b \omega+c \omega^{2}\right)$
$=-f(\alpha) f(\beta) f(\gamma)$
$\left[\because \alpha=1, \beta=\omega, \gamma=\omega^{2}\right]$