Prime Cubic Root

Algebra Level pending

Consider the general cubic polynomial with nonzero integer coefficients and constant term, namely, \(f(x) = ax^3 +bx^2 +cx +d\), where \(a>1\) is prime, \(|d|>1\) is prime, and \(f(1) = 1+d\). From the given info above, \(c = 1-a-b\). Find another expression of \(c\) that proves \(f(x)\) has exactly one rational root.

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