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MathGroup Archive 2002

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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;

 in radicals.

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[[1]]

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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