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Polynomial proof

  1. May 6, 2016 #1
    1. The problem statement, all variables and given/known data
    Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n.

    2. Relevant equations

    3. The attempt at a solution
    I tried different approaches such as analyzing the constants, f(n)-f(0)-f(1).
    How do you approach these problems in general?
  2. jcsd
  3. May 6, 2016 #2


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    What can you tell about c from knowledge of f(0) ?
  4. May 6, 2016 #3
    c=even integer
  5. May 7, 2016 #4


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    We can construct a unique parabola using 3 points. Consider the function f$$x → ax^2 + bx + c$$

    We know:
    f(0) = a1
    f(1) = a2

    a1 and a2 are even integers. You can use f(n1) for the third point. Then you have a parabola through these 3 points. Try this.
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