A Family of Cubics

Algebra Level 4

Let the general cubic polynomial with integer coefficients \(f(x)=ax^3+bx^2+cx+d\) have the trait that \(f(1)=2k\) for some integer \(k\). Does there exist a \(k\) such that \(f(x)\) has the factor \(ax^2+(c-d)x+d \)?

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