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)

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

  1. A

    0

  2. B

    1

  3. C

    -1

  4. D

    f(α)f(α)

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    Solution:

    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)=(xα)2ϕ(x). Then,

    f(x)=2(xα)ϕ(x)+(xα)2ϕ(x) f(x)f(x)=(xα)ϕ(x)2ϕ(x)+(xα)ϕ(x)

    Now,

          limxαf(x)f(x)f(x)f(x)

          =limxαf(x)f(x)f(x)f(x)f(x)f(x), where{·} denotes the

    fractional part function

    =limxαf(x)f(x)×f(x)f(x)limxαf(x)f(x)f(x)f(x)=1limxαxα2ϕ(x)+(xα)ϕ(x)2ϕ(x)+(xα)ϕ(x)(xα)ϕ(x)=10=1.

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