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Solving the Quintic: Correction to x^5 + 20 x + 32 == 0 root example
*To*: mathgroup at smc.vnet.net
*Subject*: [mg95311] Solving the Quintic: Correction to x^5 + 20 x + 32 == 0 root example
*From*: "Q.E.D." <aoe at netzero.net>
*Date*: Thu, 15 Jan 2009 06:10:40 -0500 (EST)
*Organization*: Adtech Computers, Inc.
There is an error on one of the "Solving the Quintic"
pages. It is necessary to change a minus sign to a plus
sign to get a correct N[%] value -1.363962165089939`
The change is marked below with the comment (*fixed*).
The "Solving the Quintic" page in error is
http://library.wolfram.com/examples/quintic/main.html
The page shows the following:
There are polynomials of degree greater than four which
do not factor over the rationals, but that can still be
solved in radicals. An example is the quintic:
x^5 + 20 x + 32 == 0
one of whose roots is
1/5 (-(2500 Sqrt[5] + 250 Sqrt[50 - 10 Sqrt[5]] -
750 Sqrt[50 + 10 Sqrt[5]])^(1/5) -
(-2500 Sqrt[5] + 750 Sqrt[50 - 10 Sqrt[5]] +
250 Sqrt[50 + 10 Sqrt[5]])^(1/5) +(*fixed*)
(2500 Sqrt[5] + 750 Sqrt[50 - 10 Sqrt[5]] +
250 Sqrt[50 + 10 Sqrt[5]])^(1/5) -
(2500 Sqrt[5] - 250 Sqrt[50 - 10 Sqrt[5]] +
750 Sqrt[50 + 10 Sqrt[5]])^(1/5))
The corrected result can be checked using:
N[#^5 + 20 # + 32 == 0 &[%], 100]
Can anyone who has the "Solving the Quintic" poster see
if it is correct on there?
All of the roots are given by evaluating this:
Module[{prim1, prim2, prim3, prim4, r1, r2, r3, r4},
r1 = -(2500*5^(1/2) + 250* (50 - 10*5^(1/2))^(1/2) -
750 *(50 + 10*5^(1/2))^(1/2))^(1/5);
r2 = (2500*5^(1/2) + 750 *(50 - 10*5^(1/2))^(1/2) +
250 *(50 + 10*5^(1/2))^(1/2))^(1/5);
r3 = -(-2500*5^(1/2) + 750 *(50 - 10*5^(1/2))^(1/2) +
250* (50 + 10*5^(1/2))^(1/2))^(1/5);
r4 = -(2500*5^(1/2) - 250* (50 - 10*5^(1/2))^(1/2) +
750 *(50 + 10*5^(1/2))^(1/2))^(1/5);
prim1 = (-1 - 5^(1/2) + 2^(1/2)*I*(5 - 5^(1/2))^(1/2))/4;
prim2 = (-1 + 5^(1/2) - 2^(1/2)*I*(5 + 5^(1/2))^(1/2))/4;
prim3 = (-1 + 5^(1/2) + 2^(1/2)*I*(5 + 5^(1/2))^(1/2))/4;
prim4 = (-1 - 5^(1/2) - 2^(1/2)*I*(5 - 5^(1/2))^(1/2))/4;
{x -> #} & /@ ({{r1, r2, r3, r4}.{1, 1, 1, 1},
{r1, r2, r3, r4}.{prim2, prim4, prim1, prim3},
{r1, r2, r3, r4}.{prim3, prim1, prim4, prim2},
{r1, r2, r3, r4}.{prim1, prim2, prim3, prim4},
{r1, r2, r3, r4}.{prim4, prim3, prim2, prim1}}/5)]
Check the roots using:
N[Solve[x^5 + 20 x + 32 == 0, x], 100] == N[%, 100]
The "Solving the Quintic" poster notebooks are available at
http://library.wolfram.com/infocenter/TechNotes/158/
The Radical.nb notebook is what was used here. Change
the simp function to do Expand[Simplify[%]] on its result
and discard the Discriminant definition which is built into
versions 6 & 7 to get the results in a form like what is
shown here. Also change Solve::rad to SolveQuintic::rad
in two places before evaluating all cells.
Q.E.D.
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