Let α be a root of the equation (a-c) x^2+(b-a) x+(c-b)=0 where a, b, c are distinct real numbers such that the matrix [α^2, α, 1; 1, 1, 1; a, b, c]…

Let $\displaystyle \alpha$ be a root of the equation $\displaystyle (a-c) x^2+(b-a) x+(c-b)=0$ where $\displaystyle \mathrm{a}, \mathrm{b}, \mathrm{c}$ are distinct real numbers such that the matrix $\displaystyle \left[\begin{array}{ccc}\alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{array}\right]$ is singular. Then, the value of $\displaystyle \frac{(a-c)^2}{(b-a)(c-b)}+\frac{(b-a)^2}{(a-c)(c-b)}+\frac{(c-b)^2}{(a-c)(b-a)}$ is