# 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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