Le 10/08/2010 14:17, Invisible a écrit :
> Le_Forgeron wrote:
> 
>> Solving Third & Fourth degres polynomials is known since a long time (I
>> have a book with that which was printed in the 60')
> 
> I'll bet to hell nobody actually *uses* the formulas above though. (!)
> 

Humans have a tendancy to make substitution of variables instead.

>> Now, the sad thing is it soon stop working for fifth and higher.
> 
> I thought it only stops working in terms of "elementary" functions? As
> in, there are more specialised functions you can use to go a little bit
> higher still.
> 
> Also, I thought it was a case of being impossible to write a single
> closed-form formula for the solution to an arbitrary high-order
> polynomial. Like, if the polynomial has a special form, it might still
> be solvable (possibly easily).

Yes, of course x^9 + a x^6 + b x^3 + c = 0 is just a rewrite of y^3 + a
y^2 + b y + c = 0, with y = x^3.

Solving fifth and higher requiers usually to get a first root, then
dividing the polynomial by (x-root) to reduce the degre (and start
over). Sometimes it is possible to extract a second degre polynomial
instead... that is where all the "(a+b)²=a²+2ab+b²" and the like come
handy when trying to move to the formula to a product instead of a sum)






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