Aurora Borealis!

Calculus Level 5

If \(f(x) = x^3+ax^2+bx+c\) has three distinct integral roots and given that \(f(x^2+2x+2)=0\) has no real roots. Under the given constraints we can minimise the values of \(a,\) \(b\) and \(c\). Let the minimum values of \(a,\) \(b\) and \(c\) be \(d,\) \(e\) and \(f\) respectively. Now let \(p(x) = x^3+dx^2+ex+f.\) .

Let \(p'(x)\) denote the polynomial obtained by differentiating \(p(x)\) once with respect to \(x\).

Find the value of \(k\) such that \(p'(x) = k\) has equal roots.

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