From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: cubic and quintic formulas
Date: 19 Jan 1999 17:45:26 GMT
Newsgroups: sci.math

Hub-R-ISS-1 <hopehubris@geocities.com> wrote:
>
>Oren Moses Fromberg wrote in message <36A3C589.F4EA6F24@home.com>...
>>what are the cubic and quintic formulas?
>
>I don't know what the cubic formula is, but I do know that it's really
>really REALLY nasty. So I've heard anyway.

Come now, it's not that bad. Use a linear substitution to eliminate the
quadratic term. Writing the resulting equation in the form
x^3 + 3p x + 2q = 0, the solution may be written
	cbrt(-q + sqrt(q^2+p^3)) + cbrt(-q - sqrt(q^2+p^3)).
(cbrt = "cube root of"). Is that so bad? Wait till you see the quartic! My 
favorite display of the solution is in the book "Mathematics Made Difficult".

I should remark that what's harder is to apply this formula correctly so
that it yields exactly three (not one, or nine) complex roots, and to
figure out what's going on when all three roots are real (which happens
iff  p and q are real and  q^2+p^3 <= 0 ).

>It was proven by [insert name of mathematician here] that there exists no
>quintic formula, and it was subsequently proven by [insert name of other
>mathematician] here that there is no formula that solves equations of degree
>n>4. I can't tell you anything about the proofs, though. Someone else can,
>I'm sure.

Abel and Ruffini, then Galois, showed it is impossible to express the
solution to the general polynomial of degree 5 or more with a formula
built up from the coefficients of the polynomial using only addition,
subtraction, multiplication, and the extraction of roots.(*)  There just
aren't as many numbers which can be formed using these processes as there
are solutions to polynomial equations.

(It's trivial to see that this is true for polynomials of degree n+1 if
it's true for polynomials of degree  n.)

Whether or not this is just what's to be allowed for a "formula" is a
matter of taste. Certainly from a numerical standpoint, the whole idea
of "formula" should be tossed -- it's much more stable to compute the
roots by various iterative methods.

For some common questions about polynomials you might want to start with
	index/12FXX.html

dave


(*)  '...the different branches of Arithmetic--
Ambition, Distraction, Uglification, and Derision.'
	-- Lewis Carroll, courtesy Project Gutenberg