1000 Degree Polynomial

Algebra Level 5

How many polynomials \(f(x) \) are there, which satisfy all the following conditions:

  1. \(f(x)\) is monic.
  2. \(f(x) \) has degree 1000.
  3. \(f(x) \) has real coefficients.
  4. \( f(x) \) divides \( (x-1)f(2x) \).
  5. \( f(2) \) is an integer.

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial \( x^3 + 3x - 5 \) is monic but the polynomial \( -x^4 + 2x^3 - 6 \) is not.

As stated in the problem, the coefficients are real values, not necessarily integer valued.

Clarification: The divisibility condition in 4 is about polynomials, not integer values obtained upon evaluation. For example, \(x-1\) divides \(x^2 -1\). since \(x^2-1 = (x-1)(x+1)\). \(x-2\) does not divide \(x^2-1\), even though when "\(x=3\)", 1 divides 8.

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