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

(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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