How to factor $ x^3 − 2ax −a $

Question

I have:

$$ x^3 -2k^2x - k^2 = 0 $$

Now, I want to factor it so that I can easily solve for $x$, as you can see I cannot directy use quadratic formula. AFAIK, I am expecting 3 terms here like $(a-x)(b-x)(c-x)=0$. Thanks in advance and more power.

Answer 1

Since it is in the form of a monic depressed cubic (leading coefficient=1 and no quadratic term), Cardano's method is a good way to go to find the solutions. Unfortunately, for arbitrary $k,$ the solutions aren't pretty, so the factorization is not going to be easily done.

Answer 2

You should check on cubic polynomials. Yours is the special case of a depressed cubic, the details of the solution method are given in the linked Wikipedia entry. My tame computer algebra system maxima gives the three roots, they are a veritable mess.