If `f(x)=x^2+b x^2+c x+da n df(0),f(-1)` are odd integers, prove that `f(x)=0` cannot have all integral roots.

If f(x)=x^(3)+bx^(2)+cx+d and f(0),f(-1) are odd integers,prove that f(x)=0 cannot have all integral roots.

Let f(x) be a polynomial with integral coefficients. If f(1) and f(2) both are odd integers, prove that f(x) = 0 can' t have any integral root.

Let f(x) be a quadratic polynomial with integer coefficients such that f(0) and f(1) are odd integers.then prove that the equation f(x)=0 does not have an integer solution

Let f(x)=x^2-bx+c,b is an odd positive integer. Given that f(x)=0 has two prime numbers as roots and b+c=35. If the least value of f(x) AA x in R" is "lambda," then "[|lambda/3|] is equal to (where [.] denotes greatest integer function)

If f(x) is a real valued polynomial and f (x) = 0 has real and distinct roots, show that the (f'(x)^(2)-f(x)f''(x))=0 can not have real roots.

f(x) is a cubic polynomial x^(3)+ax^(2)+bx+c such that f(x)=0 has three distinct integral roots and f(g(x))=0 does not have real roots,where g(x)=x^(2)+2x-5, the minimum value of a+b+c is

If f(x)=(a-x^(n))^((1)/(n)),a>0 and n in N, then prove that f(f(x))=x for all x.

Let f(x)=Ax^(2)+Bx+c, where A,B,C are real numbers.Prove that if f(x) is an integer whenever x is an integer,then the numbers 2A,A+B, and C are all integer. Conversely,prove that if the number 2A,A+B, and C are all integers,then f(x) is an integer whenever x is integer.