Successive Cubic Rational Root

Algebra Level pending

Let \(f(x) = ax^3 + bx^2 + cx + d\) where \(a\) and \(c\) and \(f(1) \) are successive positive integers, or \(0 < c < a=2k < f(1)\) for some integer \(k\). Find \(d\) such that \( f(x) = (ax + d)[x^2 + (c-ad)x +d]\).

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