Let `P(x) =x^2 + 1/2 x + b` and `Q(x) = x^2 + cx + d` be two polynomials with real coefficients such that `P(x)Q(x) = Q(P(x))` for all real `x`. Find all the real roots of `P(Q(x)) = 0`.

Let P(x) =x^(2)+ 1/2x + b and Q(x) = x^(2) + cx + d be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all the real roots of P(Q(x)) =0.

Let p (x) be a polynomial with real coefficient and p (x)=p'(x) =x^(2)+2x+1. Find P (1).

Let p (x) be a polynomial with real coefficient and p (x)-p'(x) =x^(2)+2x+1. Find P (-1).

Let P(x) be a polynomial of degree n with real coefficients and is non negative for all real x then P(x) + P^(1)(x)+ ...+P^(n)(x) is

Let Q(x) be a quadratic polynomial with real coefficients such that for all real x the relation, 2(1+Q( x)) =Q(x - 1) +Q(x +1) holds good. If Q(0) = 8 and Q(2)= 32 then The range of Q(x) contains 'n' distinct negative integral values then the sum of these values is

If P(x) = ax^(2) + bx + c , and Q(x) = -ax^(2) + dx + c, ax ne 0 then prove that P(x), Q(x) =0 has atleast two real roots.

Let Q(x) be a quadratic polynomial with real coefficients such that for all real x the relation, 2(1+Q(x)) =Q(x - 1) +Q(x +1) holds good. If Q(0) = 8 and Q(2) = 32 then The number of integral values of 'x' for which Q(x) gives negative real values.