Let f(x) be a polynomial satisfying (f(α))2+f′(α)2=0. Then, limx→α f(x)f′(x)f′(x)f(x) is equal to (Here [·] denotes the greatest integer function)

It is given that the polynomial f (x) satisfies the relation (f(α))2+f′(α)2=0.∴ f(α)=0=f′(α)⇒  x=a is a root of f (x) and f' (x)⇒   (x-a)2 is a factor of f (x)Let f(x)=(x−α)2ϕ(x). Then,f′(x)=2(x−α)ϕ(x)+(x−α)2ϕ′(x)∴ f(x)f′(x)=(x−α)ϕ(x)2ϕ(x)+(x−α)ϕ′(x)Now,      limx→α f(x)f′(x)f′(x)f(x)      =limx→α f(x)f′(x)f′(x)f(x)−f′(x)f(x), where{·} denotes thefractional part function=limx→α f(x)f′(x)×f′(x)f(x)−limx→α f(x)f′(x)f′(x)f(x)=1−limx→α x−α2ϕ(x)+(x−α)ϕ(x)2ϕ(x)+(x−α)ϕ′(x)(x−α)ϕ(x)=1−0=1.