What are the roots of $y^3 - ky^2 - k=0$ for an arbitrary constant $k$?
I haven't been able to find a root, so synthetic division couldn't be used. I have tried factorization as well.
2026-03-29 20:54:10.1774817650
How to determine roots to the equation $y^3 - ky^2 - k=0$
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Hint
If you apply the method for the cubic equation, you will first find $$\Delta=-k^2 \left(4 k^2+27\right) \quad < 0 \qquad \forall k$$ So, there is only one real root.
Continuing the described method, we have $p=-\frac{k^2}{3} < 0$ and $q=-\frac{2}{27} k^3-k$ and continue with the hyperbolic solution.