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KVPY2019MathematicsQuadratic Equation

Let P be a non-zero polynomial such that P 1 + x = P 1 - x for all real x and P 1 = 0 . Let m be the largest integer such that x - 1 m divides P x for all such P x . Then, m equals

Options

  1. A1
  2. B2
  3. C3
  4. D4

Correct answer

B. 2

Step-by-step solution

It is given that P is a non-zero polynomial such that, P 1 + x = P 1 - x On differentiating both sides w.r.t. x , we get P ' 1 + x = - P ' 1 - x Now on putting x = 0 , we get P ' 1 = - P ' 1 ⇒   P ' 1 = 0 i.e the polynomial P touches the X -axis at x = 1 and P 1 = 0 (given) ∴   P x = x - 1 2 Q x ∴ Largest integral value of ' m ' such that x - 1 m divides P x is 2 .

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