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MathGroup Archive 2002

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Re: Factoring a polynomial

  • To: mathgroup at smc.vnet.net
  • Subject: [mg37107] Re: Factoring a polynomial
  • From: "Allan Hayes" <hay at haystack.demon.co.uk>
  • Date: Thu, 10 Oct 2002 03:20:36 -0400 (EDT)
  • References: <ao0tp8$gsq$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Carlos,
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 (08/012/02) ago, 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 express the factors in terms of radicals

p3=ToRadicals/@p2

(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:

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
   5*(3 + 10*x*(6 + 5*x*(9 + 10*x*(3 + 7*x)))))

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 still use Re and Im

--
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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