       Re: Factoring a polynomial (2)

• To: mathgroup at smc.vnet.net
• Subject: [mg37112] Re: Factoring a polynomial (2)
• From: "Allan Hayes" <hay at haystack.demon.co.uk>
• Date: Thu, 10 Oct 2002 03:20:44 -0400 (EDT)
• References: <ao0tp8\$gsq\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Carlos,
Futher to my previous posting (which gave the code for the function FactorR
used below), here is a complete factorisation by radicals.
I also test that the product of the factors gives the original polynomial.

We want to factor the polynomial

p1 = x^6 + (9/14)*x^5 + (9/28)*x^4 + (3/35)*x^3 + (9/700)*x^2 +
(9/8750)*x +
3/87500;

We can't expect this to be easy or even possible in terms of radicals (the
general quintic is not solvable interms of radicals).
But, using the function FactorR given in my posting, Re:factoring quartic
over radicals, sent a few days ago (08/012/02) , we get

p2 = FactorR[p1, x]

(x^2 - 2*x*Root[3 + 45*#1 + 225*#1^2 + 700*#1^3 & , 1] +
Root[-3 + 225*#1^2 - 5625*#1^4 + 87500*#1^6 & , 2]^2)*
(x^2 - 2*x*Root[1827 + 65340*#1 + 974700*#1^2 + 7824000*#1^3 +
36360000*#1^4 +
100800000*#1^5 + 156800000*#1^6 & , 2] +
Root[9 - 1350*#1^2 + 84375*#1^4 - 3056250*#1^6 - 11250000*#1^8 -
984375000*#1^10 + 7656250000*#1^12 & , 3]^2)*
(x^2 - 2*x*Root[1827 + 65340*#1 + 974700*#1^2 + 7824000*#1^3 +
36360000*#1^4 +
100800000*#1^5 + 156800000*#1^6 & , 1] +
Root[9 - 1350*#1^2 + 84375*#1^4 - 3056250*#1^6 - 11250000*#1^8 -
984375000*#1^10 + 7656250000*#1^12 & , 4]^2)

Try to change the root objects to radical form:

p3 = p2 /. r_Root :> ToRadicals[r]

(3/140 + (1/140)*(13/5)^(2/3)*3^(1/3) - (1/140)*(13/5)^(1/3)*3^(2/3) -
2*(-(3/28) - (1/28)*(13/5)^(2/3)*3^(1/3) + (1/28)*(13/5)^(1/3)*3^(2/3))*x
+
x^2)*(x^2 - 2*x*Root[1827 + 65340*#1 + 974700*#1^2 + 7824000*#1^3 +
36360000*#1^4 + 100800000*#1^5 + 156800000*#1^6 & , 2] +
Root[9 - 1350*#1 + 84375*#1^2 - 3056250*#1^3 - 11250000*#1^4 -
984375000*#1^5 + 7656250000*#1^6 & , 1])*
(x^2 - 2*x*Root[1827 + 65340*#1 + 974700*#1^2 + 7824000*#1^3 +
36360000*#1^4 +
100800000*#1^5 + 156800000*#1^6 & , 1] +
Root[9 - 1350*#1 + 84375*#1^2 - 3056250*#1^3 - 11250000*#1^4 -
984375000*#1^5 + 7656250000*#1^6 & , 2])

We succeeded with the first factor:

f1 = p3[]

3/140 + (1/140)*(13/5)^(2/3)*3^(1/3) - (1/140)*(13/5)^(1/3)*3^(2/3) -
2*(-(3/28) - (1/28)*(13/5)^(2/3)*3^(1/3) + (1/28)*(13/5)^(1/3)*3^(2/3))*x
+ x^2

The product of the other two factors, in a form avoiding root objects, is
easily found by division:

q = PolynomialQuotient[p1, f1, x]

3/3500 + ((13/5)^(1/3)*3^(2/3))/3500 + (3/175 +
(1/175)*(13/5)^(1/3)*3^(2/3))*x +
(9/70 - (1/140)*(13/5)^(2/3)*3^(1/3) + (1/28)*(13/5)^(1/3)*3^(2/3))*x^2 +
(3/7 - (1/14)*(13/5)^(2/3)*3^(1/3) + (1/14)*(13/5)^(1/3)*3^(2/3))*x^3 +
x^4

Try FactorR on this

f23 = FactorR[q, x]

(x^2 + ((-(1/2))*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920] +
Im[(1/2)*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920 + (-2250 + 25*13^(2/3)*15^(1/3) -
125*13^(1/3)*15^(2/3))/8750 +
(3*(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2)/
1225000000 - (-((2*(300 + 20*13^(1/3)*15^(2/3)))/4375) +
(1/17500)*((7500 - 250*13^(2/3)*15^(1/3) +
250*13^(1/3)*15^(2/3))*
((2250 - 25*13^(2/3)*15^(1/3) + 125*13^(1/3)*15^(2/3))/4375 -
(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2/
306250000)))/(4*Sqrt[-(117/1960) -
(9/784)*(13/5)^(2/3)*3^(1/3) -
(39*(13/5)^(1/3)*3^(2/3))/3920])]])^2 -
2*x*((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 +
Re[(1/2)*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920 + (-2250 + 25*13^(2/3)*15^(1/3) -
125*13^(1/3)*15^(2/3))/8750 +
(3*(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2)/
1225000000 - (-((2*(300 + 20*13^(1/3)*15^(2/3)))/4375) +
(1/17500)*((7500 - 250*13^(2/3)*15^(1/3) +
250*13^(1/3)*15^(2/3))*
((2250 - 25*13^(2/3)*15^(1/3) + 125*13^(1/3)*15^(2/3))/4375 -
(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2/
306250000)))/(4*Sqrt[-(117/1960) -
(9/784)*(13/5)^(2/3)*3^(1/3) -
(39*(13/5)^(1/3)*3^(2/3))/3920])]]) +
((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 +
Re[(1/2)*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920 + (-2250 + 25*13^(2/3)*15^(1/3) -
125*13^(1/3)*15^(2/3))/8750 +
(3*(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2)/
1225000000 - (-((2*(300 + 20*13^(1/3)*15^(2/3)))/4375) +
(1/17500)*((7500 - 250*13^(2/3)*15^(1/3) +
250*13^(1/3)*15^(2/3))*
((2250 - 25*13^(2/3)*15^(1/3) + 125*13^(1/3)*15^(2/3))/4375 -
(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2/
306250000)))/(4*Sqrt[-(117/1960) -
(9/784)*(13/5)^(2/3)*3^(1/3) -
(39*(13/5)^(1/3)*3^(2/3))/3920])]])^2)*
(x^2 + ((1/2)*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920] +
Im[(-(1/2))*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920 + (-2250 + 25*13^(2/3)*15^(1/3) -
125*13^(1/3)*15^(2/3))/8750 +
(3*(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2)/
1225000000 + (-((2*(300 + 20*13^(1/3)*15^(2/3)))/4375) +
(1/17500)*((7500 - 250*13^(2/3)*15^(1/3) +
250*13^(1/3)*15^(2/3))*
((2250 - 25*13^(2/3)*15^(1/3) + 125*13^(1/3)*15^(2/3))/4375 -
(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2/
306250000)))/(4*Sqrt[-(117/1960) -
(9/784)*(13/5)^(2/3)*3^(1/3) -
(39*(13/5)^(1/3)*3^(2/3))/3920])]])^2 -
2*x*((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 +
Re[(-(1/2))*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920 + (-2250 + 25*13^(2/3)*15^(1/3) -
125*13^(1/3)*15^(2/3))/8750 +
(3*(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2)/
1225000000 + (-((2*(300 + 20*13^(1/3)*15^(2/3)))/4375) +
(1/17500)*((7500 - 250*13^(2/3)*15^(1/3) +
250*13^(1/3)*15^(2/3))*
((2250 - 25*13^(2/3)*15^(1/3) + 125*13^(1/3)*15^(2/3))/4375 -
(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2/
306250000)))/(4*Sqrt[-(117/1960) -
(9/784)*(13/5)^(2/3)*3^(1/3) -
(39*(13/5)^(1/3)*3^(2/3))/3920])]]) +
((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 +
Re[(-(1/2))*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920 + (-2250 + 25*13^(2/3)*15^(1/3) -
125*13^(1/3)*15^(2/3))/8750 +
(3*(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2)/
1225000000 + (-((2*(300 + 20*13^(1/3)*15^(2/3)))/4375) +
(1/17500)*((7500 - 250*13^(2/3)*15^(1/3) +
250*13^(1/3)*15^(2/3))*
((2250 - 25*13^(2/3)*15^(1/3) + 125*13^(1/3)*15^(2/3))/4375 -
(7500 - 250*13^(2/3)*15^(1/3) + 250*13^(1/3)*15^(2/3))^2/
306250000)))/(4*Sqrt[-(117/1960) -
(9/784)*(13/5)^(2/3)*3^(1/3) -
(39*(13/5)^(1/3)*3^(2/3))/3920])]])^2)

We try to get rid of the parts Re[.] and Im[.]:,

f231 = f23 /. z:(_Re | _Im) :> ToRadicals[FullSimplify[z]]

(((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 -
(1/280)*Sqrt[3*(-390 + 13*13^(2/3)*15^(1/3) + 15*13^(1/3)*15^(2/3))])^2
+
((1/2)*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920] +
(1/280)*Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)])^2 -
2*((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 -
(1/280)*Sqrt[3*(-390 + 13*13^(2/3)*15^(1/3) + 15*13^(1/3)*15^(2/3))])*x
+
x^2)*(((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 +
(1/280)*Sqrt[3*(-390 + 13*13^(2/3)*15^(1/3) + 15*13^(1/3)*15^(2/3))])^2
+
((-(1/2))*Sqrt[117/1960 + (9/784)*(13/5)^(2/3)*3^(1/3) +
(39*(13/5)^(1/3)*3^(2/3))/3920] +
(1/280)*Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)])^2 -
2*((-7500 + 250*13^(2/3)*15^(1/3) - 250*13^(1/3)*15^(2/3))/70000 +
(1/280)*Sqrt[3*(-390 + 13*13^(2/3)*15^(1/3) + 15*13^(1/3)*15^(2/3))])*x
+
x^2)

We now have the ramaining two factors in radical form, but a little
simplification helps:

f232 = f231 /. (n_)?NumericQ :> Simplify[n]

((1/78400)*(30 - 13^(2/3)*15^(1/3) + 13^(1/3)*15^(2/3) +
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)])^2 +
(1/78400)*(Sqrt[1170 + 45*13^(2/3)*15^(1/3) + 39*13^(1/3)*15^(2/3)] +
Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)])^2 -
(1/140)*(-30 + 13^(2/3)*15^(1/3) - 13^(1/3)*15^(2/3) -
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)])*x + x^2)*
((1/78400)*(-30 + 13^(2/3)*15^(1/3) - 13^(1/3)*15^(2/3) +
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)])^2 +
(1/78400)*(Sqrt[1170 + 45*13^(2/3)*15^(1/3) + 39*13^(1/3)*15^(2/3)] -
Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)])^2 -
(1/140)*(-30 + 13^(2/3)*15^(1/3) - 13^(1/3)*15^(2/3) +
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)])*x + x^2)

TEST

Test if the product of the factors is equal to  p1:

prd1 = Collect[Expand[f232*f1], x]

172077/3841600000 - (4959*(13/5)^(2/3)*3^(1/3))/768320000 +
(117*(13/5)^(1/3)*3^(2/3))/27440000 - (1/1920800000)*
(3*Sqrt[1170 + 45*13^(2/3)*15^(1/3) + 39*13^(1/3)*15^(2/3)]*
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)]*
Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)]) +
(1/3841600000)*((13/5)^(1/3)*3^(2/3)*Sqrt[1170 + 45*13^(2/3)*15^(1/3) +
39*13^(1/3)*15^(2/3)]*Sqrt[-1170 + 39*13^(2/3)*15^(1/3) +
45*13^(1/3)*15^(2/3)]*Sqrt[1170 + 73*13^(2/3)*15^(1/3) +
67*13^(1/3)*15^(2/3)]) +
(491193/384160000 - (7731*(13/5)^(2/3)*3^(1/3))/76832000 +
(117*(13/5)^(1/3)*3^(2/3))/2744000 - (1/384160000)*
(9*Sqrt[1170 + 45*13^(2/3)*15^(1/3) + 39*13^(1/3)*15^(2/3)]*
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)]*
Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)]) -
(1/384160000)*((13/5)^(2/3)*3^(1/3)*Sqrt[1170 + 45*13^(2/3)*15^(1/3) +
39*13^(1/3)*15^(2/3)]*Sqrt[-1170 + 39*13^(2/3)*15^(1/3) +
45*13^(1/3)*15^(2/3)]*Sqrt[1170 + 73*13^(2/3)*15^(1/3) +
67*13^(1/3)*15^(2/3)]) + (1/192080000)*((13/5)^(1/3)*3^(2/3)*
Sqrt[1170 + 45*13^(2/3)*15^(1/3) + 39*13^(1/3)*15^(2/3)]*
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)]*
Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)]))*x +
(1087173/76832000 - (12771*(13/5)^(2/3)*3^(1/3))/30732800 +
(5967*(13/5)^(1/3)*3^(2/3))/30732800 - (1/76832000)*
(9*Sqrt[1170 + 45*13^(2/3)*15^(1/3) + 39*13^(1/3)*15^(2/3)]*
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)]*
Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)]) -
(1/153664000)*(3*(13/5)^(2/3)*3^(1/3)*Sqrt[1170 + 45*13^(2/3)*15^(1/3) +
39*13^(1/3)*15^(2/3)]*Sqrt[-1170 + 39*13^(2/3)*15^(1/3) +
45*13^(1/3)*15^(2/3)]*Sqrt[1170 + 73*13^(2/3)*15^(1/3) +
67*13^(1/3)*15^(2/3)]) + (1/153664000)*(3*(13/5)^(1/3)*3^(2/3)*
Sqrt[1170 + 45*13^(2/3)*15^(1/3) + 39*13^(1/3)*15^(2/3)]*
Sqrt[-1170 + 39*13^(2/3)*15^(1/3) + 45*13^(1/3)*15^(2/3)]*
Sqrt[1170 + 73*13^(2/3)*15^(1/3) + 67*13^(1/3)*15^(2/3)]))*x^2 +
(23871/274400 - (99*(13/5)^(2/3)*3^(1/3))/109760 +
(663*(13/5)^(1/3)*3^(2/3))/
548800 - (1/2744000)*(Sqrt[1170 + 45*13^(2/3)*15^(1/3) +
39*13^(1/3)*15^(2/3)]*Sqrt[-1170 + 39*13^(2/3)*15^(1/3) +
45*13^(1/3)*15^(2/3)]*Sqrt[1170 + 73*13^(2/3)*15^(1/3) +
67*13^(1/3)*15^(2/3)]))*x^3 + (9*x^4)/28 + (9*x^5)/14 + x^6

prd1 /. (n_)?NumericQ :> ToRadicals[FullSimplify[n]]

172077/3841600000 - (4959*(13/5)^(2/3)*3^(1/3))/768320000 +
(117*(13/5)^(1/3)*3^(2/3))/27440000 -
(9*(234 + 221*(13/5)^(1/3)*3^(2/3) - 33*13^(2/3)*15^(1/3)))/384160000 +
(117*(-165 + 17*13^(2/3)*15^(1/3) + 6*13^(1/3)*15^(2/3)))/3841600000 +
(9*x)/8750 + (9*x^2)/700 + (3*x^3)/35 + (9*x^4)/28 + (9*x^5)/14 + x^6

Together[%]

(3 + 90*x + 1125*x^2 + 7500*x^3 + 28125*x^4 + 56250*x^5 + 87500*x^6)/87500

Apart[%]

3/87500 + (9*x)/8750 + (9*x^2)/700 + (3*x^3)/35 + (9*x^4)/28 + (9*x^5)/14 +
x^6

This is p1:

p1

3/87500 + (9*x)/8750 + (9*x^2)/700 + (3*x^3)/35 + (9*x^4)/28 + (9*x^5)/14 +
x^6

------------------
It is ususlly better to try to reduce a difference to zero than to reduce
one form
to another

tst1 = Collect[Expand[f232*f1 - p1], x]

tst2 = tst1 /. (n_)?NumericQ :> ToRadicals[FullSimplify[n]]

8073/768320000 - (4959*(13/5)^(2/3)*3^(1/3))/768320000 +
(117*(13/5)^(1/3)*3^(2/3))/27440000 -
(9*(234 + 221*(13/5)^(1/3)*3^(2/3) - 33*13^(2/3)*15^(1/3)))/384160000 +
(117*(-165 + 17*13^(2/3)*15^(1/3) + 6*13^(1/3)*15^(2/3)))/3841600000

Together[%]

0

--
Allan

---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565

"Carlos Felippa" <carlos at colorado.edu> wrote in message
news:ao0tp8\$gsq\$1 at smc.vnet.net...
> Can Mathematica factor the polynomial
>
> p1=x^6+9/14*x^5+9/28*x^4+3/35*x^3+9/700*x^2+9/8750*x+3/87500;
>
> without a priori knowledge of the Extension field?
>

```

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