Roots of a cubic polynomial

Number Theory Level pending

Consider the cubic polynomial

\[\large p(x)=ax^3+ bx^2 + cx + d\]

where \(a\), \(b\), \(c\), and \(d\) are integers such that \(ad\) is odd and \(bc\) is even and not all roots of \(p(x)\) can be rational.

Is this possible? If yes, prove it. If no, why?


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