I Am Rational (Birthday Quartic)

Algebra Level pending

Let your birthday be expressed as mm/dd/yy where the integer \(M\) is the digits mm, \(D\) is the digits dd, and \(Y\) is the digits yy. Now let \(f(x) = Mx^4 + bx^3 + Dx^2 +Y\). Find \(b\) such that \(f(x)\) has a rational factor \(g(x)\) where \(f(x)/g(x)\) has middle terms \((D+Y)x^2 -Yx\). For example, the middle terms of the factor \((qx^3 + rx^2 + sx +t)\) are \(rx^2 + sx\).

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