If $a, b, c$ are distinct and the roots of $(b-c)x^2 + (c-a)x + (a-b) = 0$ are equal,then $a, b$ and $c$ are in

If the roots of the equation $(p-3)x^2 + 2(p-3)x + 2p-5 = 0$ are real and distinct for $\alpha < p < \beta$ and $(\beta - \alpha)$ is maximum,then the extreme value of the quadratic expression $-(\alpha + \beta)x^2 + \alpha \beta x + (\alpha - \beta)$ is

The number of distinct real roots of the equation $x^{5}(x^{3}-x^{2}-x+1)+x(3x^{3}-4x^{2}-2x+4)-1=0$ is

If $x^2 + x + 1$ is a factor of $ax^3 + bx^2 + cx + d$,then what is the real root of $ax^3 + bx^2 + cx + d = 0$?

If $\alpha, \beta$ are the irrational roots of the equation $3p^2x^3 + px^2 + qx + 3 = 0$ when $p = 1$ and $q = -7$,then $|\alpha - \beta| = $

If the equation $x^5-3x^4-5x^3+27x^2-32x+12=0$ has repeated roots,then the prime number that divides the non-repeated root of this equation is