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| ``Re: undefined roots of a cubic?''
by djao on 2004-05-14 15:47:30 |
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| You are right of course about the dividing by 0. The formula results in a term of 0/0 whenever b = a^2/3. If you are looking for restrictions on the constants then the previous sentence gives you the list of restrictions.
The 0/0 term is not "really" there of course -- it can be simplified out of the formula, in much the same way that ((x+h)^2-x^2)/h simplifies to 2x in calculus when h->0. In this case the formula is much more complicated than ((x+h)^2-x^2)/h and it is not immediately obvious (not even to me) how to go about the simplification.
If you look in the "Galois theoretic derivation of the cubic formula" which is attached to this entry, you will find there an alternative formula which never encounters division by 0. I will shortly undertake to replace the formulas on this page with the formulas from that page, because clearly it is preferable to have formulas that avoid division by 0.
In case you are wondering where I got the formulas, I simply copied and pasted the mathematica output of Solve[x^3 + a*x^2 + b*x + c == 0,x]. The formulas are not really intended for human use; in fact, I had expected their presence out in the open to discourage people from using them owing to their sheer bulk.
I suspect that the quartic formula page suffers from similar problems, but in that case the formulas really are much too unwieldy to consider fixing.
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