Let `P(x) = x^(2) + ax + b` be a quadratic polynomial with real coefficients. Suppose there are real numbers `s ne t` such that P(s) = t and P(t) = s. Prove that b - st is a root of the equation `x^(2) + ax + b st =0`.

Let a,b,c be real numbers in G.P. such that a and c are positive , then the roots of the equation ax^(2) +bx+c=0

If p , q , ra n ds are real numbers such that p r=2(q+s), then show that at least one of the equations x^2+p x+q=0 and x^2+r x+s=0 has real roots.

Let f(x) = x^4 + ax^3 + bx^2 + cx + d be a polynomial with real coefficients and real roots. If |f(i)|=1where i=sqrt(-1) , then the value of a +b+c+d is

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 f (x) =x ^(2) + ax +b and g (x) =x ^(2) +cx+d be two quadratic polynomials with real coefficients and satisfy ac =2 (b+d). Then which of the following is (are) correct ?

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 quadratic polynomical with real coefficient such tht for all real x the relation 2(1 + P(x)) = P(x - 1) + P(x + 1) holds. If P(0) = 8 and P(2) = 32 then Sum of all coefficients of P(x) can not be

Let P(x) be quadratic polynomical with real coefficient such tht for all real x(1) the relation 2(1 + P(x)) = P(x - 1) + P(x + 1) holds. If P(0) = 8 and P(2) = 32 then If the range of P(x) is [m, oo) then 'm' is less then

If a and b(!=0) are the roots of the equation x^2+ax+b=0 then the least value of x^2+ax+b is

If p + iq be one of the roots of the equation x^(3) +ax +b=0 ,then 2p is one of the roots of the equation