How Many Even Coefficients?

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Let \(f(x)\) be a \(2014\) degree polynomial whose coefficients are all even integers. Suppose \(f(x)\) can be written as \(f(x)= g(x) h(x),\) where \(g(x)\) and \(h(x)\) again are non-constant polynomials with integer coefficients, both having degree \(\dfrac{2014}{2}\). Let \(N_g\) denote the number of even coefficients of \(g(x),\) and let \(N_h\) denote the number of even coefficients of \(h(x).\) As \(f(x)\) ranges over all such irreducible \(2014\) degree polynomials in \(\mathbb{Z}[x],\) find the minimum possible value of \(N_g + N_h-1.\)

Details and assumptions

  • As an explicit example, the polynomial \(x^2+2x+1\) has \(1\) even coefficient (coefficient of \(x\)).

  • Note that in a polynomial with degree \(n,\) the coefficients of \(x^k\) for all \(k>n\) are zero. These coefficients aren't counted in the count of even coefficients.


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