If a polynomial $P(x)$ of degree $4$ is given by $P(x) = 2x^4 + ax^3 + bx^2 + cx + d$ such that $P(1) = 4, P(2) = 7, P(3) = 12$,and $P(4) = 19$,then find the value of $P(5)$.

If the equation whose roots are $p$ times the roots of the equation $x^4-2ax^3+4bx^2+8ax+16=0$ is a reciprocal equation,then $|p|=$ :

Let $x, y, z$ be positive real numbers. Which of the following conditions imply $x=y=z$?
$I.$ $x^3+y^3+z^3=3xyz$
$II.$ $x^3+y^2z+yz^2=3xyz$
$III.$ $x^3+y^2z+z^2x=3xyz$
$IV.$ $(x+y+z)^3=27xyz$

The polynomial equation of degree $5$ whose roots are the translates of the roots of $x^5-2x^4+3x^3-4x^2+5x-6=0$ by $-2$ is:

If the roots of the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{5}{2}$ are $p$ and $q$ $(p > q)$ and the roots of the equation $(p+q)x^4 - pqx^2 + \frac{p}{q} = 0$ are $\alpha, \beta, \gamma, \delta$,then $(\Sigma \alpha)^2 - \Sigma \alpha \beta + \alpha \beta \gamma \delta = $

For what condition will the expression $a^2x^2 + bx + 1$ be positive for all $x \in R$?