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Factorization of the Coefficient of x 11
Sean Trendell
10/16/2017

var ( 'x , a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , p1 , p2 , p3 , p4 , q1 , q2 , q3 , q4 ' )
w0 = SR . w i l d ( 0 )
s a g e _ s e r v e r .MAX_STDOUT_SIZE=250000
s a g e _ s e r v e r .MAX_OUTPUT=250000
#This d e f i n e s our v a r i a b l e s , s e t s up our w i l d c a r d and i n c r e a s e s max output f o r \
t h e program .
(x, a1, a2, a3, a4, b1, b2, b3, b4, p1, p2, p3, p4, q1, q2, q3, q4)
E=(x - a1 * a2 * a3 * a4 ) * ( x - a1 * a2 * a3 *b4 ) * ( x - a1 * a2 *b3* a4 ) * ( x - a1 * a2 *b3*b4 ) * ( x - a1 *b2* a3 *\
a4 ) * ( x - a1 *b2* a3 *b4 ) * ( x - a1 *b2*b3* a4 ) * ( x - a1 *b2*b3*b4 ) * ( x - b1* a2 * a3 * a4 ) * ( x - b1* a2\
* a3 *b4 ) * ( x - b1* a2 *b3* a4 ) * ( x - b1* a2 *b3*b4 ) * ( x - b1*b2* a3 * a4 ) * ( x - b1*b2* a3 *b4 ) * ( x - \
b1*b2*b3* a4 ) * ( x - b1*b2*b3*b4 )
F=E . expand ( )
#M u l t i p l i c a t i o n o f t h e c h a r a c t e r i s t i c f u n c t i o n , t h i s i s not show do t o l e n g t h .
g11=-F . c o e f f i c i e n t ( x , 1 1 )
g11
#Ga th er in g o f t h e c o e f f i c i e n t f o r x^11 from our e x p a n s i o n o f t h e c h a r a c t e r i s t i c \
f u n c t i o n . We t a k e t h e n e g a t i v e b e c a u s e we h a n d l e t h e s i g n i n t h e f i n a l form\
of the c h a r a c t e r i s t i c .
a1^3*a2^3*a3^4*a4^5*b1^2*b2^2*b3 + a1^2*a2^3*a3^4*a4^5*b1^3*b2^2*b3 +
a1^3*a2^2*a3^4*a4^5*b1^2*b2^3*b3 + a1^2*a2^2*a3^4*a4^5*b1^3*b2^3*b3 +
a1^3*a2^4*a3^3*a4^5*b1^2*b2*b3^2 + a1^2*a2^4*a3^3*a4^5*b1^3*b2*b3^2 +
a1^4*a2^3*a3^3*a4^5*b1*b2^2*b3^2 + 4*a1^3*a2^3*a3^3*a4^5*b1^2*b2^2*b3^2 +
4*a1^2*a2^3*a3^3*a4^5*b1^3*b2^2*b3^2 + a1*a2^3*a3^3*a4^5*b1^4*b2^2*b3^2 +
a1^4*a2^2*a3^3*a4^5*b1*b2^3*b3^2 + 4*a1^3*a2^2*a3^3*a4^5*b1^2*b2^3*b3^2 +
4*a1^2*a2^2*a3^3*a4^5*b1^3*b2^3*b3^2 + a1*a2^2*a3^3*a4^5*b1^4*b2^3*b3^2 +
a1^3*a2*a3^3*a4^5*b1^2*b2^4*b3^2 + a1^2*a2*a3^3*a4^5*b1^3*b2^4*b3^2 +
a1^3*a2^4*a3^2*a4^5*b1^2*b2*b3^3 + a1^2*a2^4*a3^2*a4^5*b1^3*b2*b3^3 +
a1^4*a2^3*a3^2*a4^5*b1*b2^2*b3^3 + 4*a1^3*a2^3*a3^2*a4^5*b1^2*b2^2*b3^3 +
4*a1^2*a2^3*a3^2*a4^5*b1^3*b2^2*b3^3 + a1*a2^3*a3^2*a4^5*b1^4*b2^2*b3^3 +
a1^4*a2^2*a3^2*a4^5*b1*b2^3*b3^3 + 4*a1^3*a2^2*a3^2*a4^5*b1^2*b2^3*b3^3 +
4*a1^2*a2^2*a3^2*a4^5*b1^3*b2^3*b3^3 + a1*a2^2*a3^2*a4^5*b1^4*b2^3*b3^3 +
a1^3*a2*a3^2*a4^5*b1^2*b2^4*b3^3 + a1^2*a2*a3^2*a4^5*b1^3*b2^4*b3^3 +
a1^3*a2^3*a3*a4^5*b1^2*b2^2*b3^4 + a1^2*a2^3*a3*a4^5*b1^3*b2^2*b3^4 +
a1^3*a2^2*a3*a4^5*b1^2*b2^3*b3^4 + a1^2*a2^2*a3*a4^5*b1^3*b2^3*b3^4 +
a1^3*a2^3*a3^5*a4^4*b1^2*b2^2*b4 + a1^2*a2^3*a3^5*a4^4*b1^3*b2^2*b4 +
a1^3*a2^2*a3^5*a4^4*b1^2*b2^3*b4 + a1^2*a2^2*a3^5*a4^4*b1^3*b2^3*b4 +
a1^4*a2^4*a3^4*a4^4*b1*b2*b3*b4 + 3*a1^3*a2^4*a3^4*a4^4*b1^2*b2*b3*b4 +
3*a1^2*a2^4*a3^4*a4^4*b1^3*b2*b3*b4 + a1*a2^4*a3^4*a4^4*b1^4*b2*b3*b4 +
3*a1^4*a2^3*a3^4*a4^4*b1*b2^2*b3*b4 + 10*a1^3*a2^3*a3^4*a4^4*b1^2*b2^2*b3*b4 +
10*a1^2*a2^3*a3^4*a4^4*b1^3*b2^2*b3*b4 + 3*a1*a2^3*a3^4*a4^4*b1^4*b2^2*b3*b4 +
3*a1^4*a2^2*a3^4*a4^4*b1*b2^3*b3*b4 + 10*a1^3*a2^2*a3^4*a4^4*b1^2*b2^3*b3*b4 +
10*a1^2*a2^2*a3^4*a4^4*b1^3*b2^3*b3*b4 + 3*a1*a2^2*a3^4*a4^4*b1^4*b2^3*b3*b4 +
a1^4*a2*a3^4*a4^4*b1*b2^4*b3*b4 + 3*a1^3*a2*a3^4*a4^4*b1^2*b2^4*b3*b4 +
3*a1^2*a2*a3^4*a4^4*b1^3*b2^4*b3*b4 + a1*a2*a3^4*a4^4*b1^4*b2^4*b3*b4 +
a1^3*a2^5*a3^3*a4^4*b1^2*b3^2*b4 + a1^2*a2^5*a3^3*a4^4*b1^3*b3^2*b4 +
3*a1^4*a2^4*a3^3*a4^4*b1*b2*b3^2*b4 + 10*a1^3*a2^4*a3^3*a4^4*b1^2*b2*b3^2*b4 +

1

10*a1^2*a2^4*a3^3*a4^4*b1^3*b2*b3^2*b4 + 3*a1*a2^4*a3^3*a4^4*b1^4*b2*b3^2*b4 +
a1^5*a2^3*a3^3*a4^4*b2^2*b3^2*b4 + 10*a1^4*a2^3*a3^3*a4^4*b1*b2^2*b3^2*b4 +
27*a1^3*a2^3*a3^3*a4^4*b1^2*b2^2*b3^2*b4 + 27*a1^2*a2^3*a3^3*a4^4*b1^3*b2^2*b3^2*b4 +
10*a1*a2^3*a3^3*a4^4*b1^4*b2^2*b3^2*b4 + a2^3*a3^3*a4^4*b1^5*b2^2*b3^2*b4 +
a1^5*a2^2*a3^3*a4^4*b2^3*b3^2*b4 + 10*a1^4*a2^2*a3^3*a4^4*b1*b2^3*b3^2*b4 +
27*a1^3*a2^2*a3^3*a4^4*b1^2*b2^3*b3^2*b4 + 27*a1^2*a2^2*a3^3*a4^4*b1^3*b2^3*b3^2*b4 +
10*a1*a2^2*a3^3*a4^4*b1^4*b2^3*b3^2*b4 + a2^2*a3^3*a4^4*b1^5*b2^3*b3^2*b4 +
3*a1^4*a2*a3^3*a4^4*b1*b2^4*b3^2*b4 + 10*a1^3*a2*a3^3*a4^4*b1^2*b2^4*b3^2*b4 +
10*a1^2*a2*a3^3*a4^4*b1^3*b2^4*b3^2*b4 + 3*a1*a2*a3^3*a4^4*b1^4*b2^4*b3^2*b4 +
a1^3*a3^3*a4^4*b1^2*b2^5*b3^2*b4 + a1^2*a3^3*a4^4*b1^3*b2^5*b3^2*b4 +
a1^3*a2^5*a3^2*a4^4*b1^2*b3^3*b4 + a1^2*a2^5*a3^2*a4^4*b1^3*b3^3*b4 +
3*a1^4*a2^4*a3^2*a4^4*b1*b2*b3^3*b4 + 10*a1^3*a2^4*a3^2*a4^4*b1^2*b2*b3^3*b4 +
10*a1^2*a2^4*a3^2*a4^4*b1^3*b2*b3^3*b4 + 3*a1*a2^4*a3^2*a4^4*b1^4*b2*b3^3*b4 +
a1^5*a2^3*a3^2*a4^4*b2^2*b3^3*b4 + 10*a1^4*a2^3*a3^2*a4^4*b1*b2^2*b3^3*b4 +
27*a1^3*a2^3*a3^2*a4^4*b1^2*b2^2*b3^3*b4 + 27*a1^2*a2^3*a3^2*a4^4*b1^3*b2^2*b3^3*b4 +
10*a1*a2^3*a3^2*a4^4*b1^4*b2^2*b3^3*b4 + a2^3*a3^2*a4^4*b1^5*b2^2*b3^3*b4 +
a1^5*a2^2*a3^2*a4^4*b2^3*b3^3*b4 + 10*a1^4*a2^2*a3^2*a4^4*b1*b2^3*b3^3*b4 +
27*a1^3*a2^2*a3^2*a4^4*b1^2*b2^3*b3^3*b4 + 27*a1^2*a2^2*a3^2*a4^4*b1^3*b2^3*b3^3*b4 +
10*a1*a2^2*a3^2*a4^4*b1^4*b2^3*b3^3*b4 + a2^2*a3^2*a4^4*b1^5*b2^3*b3^3*b4 +
3*a1^4*a2*a3^2*a4^4*b1*b2^4*b3^3*b4 + 10*a1^3*a2*a3^2*a4^4*b1^2*b2^4*b3^3*b4 +
10*a1^2*a2*a3^2*a4^4*b1^3*b2^4*b3^3*b4 + 3*a1*a2*a3^2*a4^4*b1^4*b2^4*b3^3*b4 +
a1^3*a3^2*a4^4*b1^2*b2^5*b3^3*b4 + a1^2*a3^2*a4^4*b1^3*b2^5*b3^3*b4 +
a1^4*a2^4*a3*a4^4*b1*b2*b3^4*b4 + 3*a1^3*a2^4*a3*a4^4*b1^2*b2*b3^4*b4 +
3*a1^2*a2^4*a3*a4^4*b1^3*b2*b3^4*b4 + a1*a2^4*a3*a4^4*b1^4*b2*b3^4*b4 +
3*a1^4*a2^3*a3*a4^4*b1*b2^2*b3^4*b4 + 10*a1^3*a2^3*a3*a4^4*b1^2*b2^2*b3^4*b4 +
10*a1^2*a2^3*a3*a4^4*b1^3*b2^2*b3^4*b4 + 3*a1*a2^3*a3*a4^4*b1^4*b2^2*b3^4*b4 +
3*a1^4*a2^2*a3*a4^4*b1*b2^3*b3^4*b4 + 10*a1^3*a2^2*a3*a4^4*b1^2*b2^3*b3^4*b4 +
10*a1^2*a2^2*a3*a4^4*b1^3*b2^3*b3^4*b4 + 3*a1*a2^2*a3*a4^4*b1^4*b2^3*b3^4*b4 +
a1^4*a2*a3*a4^4*b1*b2^4*b3^4*b4 + 3*a1^3*a2*a3*a4^4*b1^2*b2^4*b3^4*b4 +
3*a1^2*a2*a3*a4^4*b1^3*b2^4*b3^4*b4 + a1*a2*a3*a4^4*b1^4*b2^4*b3^4*b4 +
a1^3*a2^3*a4^4*b1^2*b2^2*b3^5*b4 + a1^2*a2^3*a4^4*b1^3*b2^2*b3^5*b4 +
a1^3*a2^2*a4^4*b1^2*b2^3*b3^5*b4 + a1^2*a2^2*a4^4*b1^3*b2^3*b3^5*b4 +
a1^3*a2^4*a3^5*a4^3*b1^2*b2*b4^2 + a1^2*a2^4*a3^5*a4^3*b1^3*b2*b4^2 +
a1^4*a2^3*a3^5*a4^3*b1*b2^2*b4^2 + 4*a1^3*a2^3*a3^5*a4^3*b1^2*b2^2*b4^2 +
4*a1^2*a2^3*a3^5*a4^3*b1^3*b2^2*b4^2 + a1*a2^3*a3^5*a4^3*b1^4*b2^2*b4^2 +
a1^4*a2^2*a3^5*a4^3*b1*b2^3*b4^2 + 4*a1^3*a2^2*a3^5*a4^3*b1^2*b2^3*b4^2 +
4*a1^2*a2^2*a3^5*a4^3*b1^3*b2^3*b4^2 + a1*a2^2*a3^5*a4^3*b1^4*b2^3*b4^2 +
a1^3*a2*a3^5*a4^3*b1^2*b2^4*b4^2 + a1^2*a2*a3^5*a4^3*b1^3*b2^4*b4^2 +
a1^3*a2^5*a3^4*a4^3*b1^2*b3*b4^2 + a1^2*a2^5*a3^4*a4^3*b1^3*b3*b4^2 +
3*a1^4*a2^4*a3^4*a4^3*b1*b2*b3*b4^2 + 10*a1^3*a2^4*a3^4*a4^3*b1^2*b2*b3*b4^2 +
10*a1^2*a2^4*a3^4*a4^3*b1^3*b2*b3*b4^2 + 3*a1*a2^4*a3^4*a4^3*b1^4*b2*b3*b4^2 +
a1^5*a2^3*a3^4*a4^3*b2^2*b3*b4^2 + 10*a1^4*a2^3*a3^4*a4^3*b1*b2^2*b3*b4^2 +
27*a1^3*a2^3*a3^4*a4^3*b1^2*b2^2*b3*b4^2 + 27*a1^2*a2^3*a3^4*a4^3*b1^3*b2^2*b3*b4^2 +
10*a1*a2^3*a3^4*a4^3*b1^4*b2^2*b3*b4^2 + a2^3*a3^4*a4^3*b1^5*b2^2*b3*b4^2 +
a1^5*a2^2*a3^4*a4^3*b2^3*b3*b4^2 + 10*a1^4*a2^2*a3^4*a4^3*b1*b2^3*b3*b4^2 +
27*a1^3*a2^2*a3^4*a4^3*b1^2*b2^3*b3*b4^2 + 27*a1^2*a2^2*a3^4*a4^3*b1^3*b2^3*b3*b4^2 +
10*a1*a2^2*a3^4*a4^3*b1^4*b2^3*b3*b4^2 + a2^2*a3^4*a4^3*b1^5*b2^3*b3*b4^2 +
3*a1^4*a2*a3^4*a4^3*b1*b2^4*b3*b4^2 + 10*a1^3*a2*a3^4*a4^3*b1^2*b2^4*b3*b4^2 +
10*a1^2*a2*a3^4*a4^3*b1^3*b2^4*b3*b4^2 + 3*a1*a2*a3^4*a4^3*b1^4*b2^4*b3*b4^2 +
a1^3*a3^4*a4^3*b1^2*b2^5*b3*b4^2 + a1^2*a3^4*a4^3*b1^3*b2^5*b3*b4^2 +
a1^4*a2^5*a3^3*a4^3*b1*b3^2*b4^2 + 4*a1^3*a2^5*a3^3*a4^3*b1^2*b3^2*b4^2 +
4*a1^2*a2^5*a3^3*a4^3*b1^3*b3^2*b4^2 + a1*a2^5*a3^3*a4^3*b1^4*b3^2*b4^2 +
a1^5*a2^4*a3^3*a4^3*b2*b3^2*b4^2 + 10*a1^4*a2^4*a3^3*a4^3*b1*b2*b3^2*b4^2 +
27*a1^3*a2^4*a3^3*a4^3*b1^2*b2*b3^2*b4^2 + 27*a1^2*a2^4*a3^3*a4^3*b1^3*b2*b3^2*b4^2 +
10*a1*a2^4*a3^3*a4^3*b1^4*b2*b3^2*b4^2 + a2^4*a3^3*a4^3*b1^5*b2*b3^2*b4^2 +
4*a1^5*a2^3*a3^3*a4^3*b2^2*b3^2*b4^2 + 27*a1^4*a2^3*a3^3*a4^3*b1*b2^2*b3^2*b4^2 +
64*a1^3*a2^3*a3^3*a4^3*b1^2*b2^2*b3^2*b4^2 + 64*a1^2*a2^3*a3^3*a4^3*b1^3*b2^2*b3^2*b4^2 +
27*a1*a2^3*a3^3*a4^3*b1^4*b2^2*b3^2*b4^2 + 4*a2^3*a3^3*a4^3*b1^5*b2^2*b3^2*b4^2 +
4*a1^5*a2^2*a3^3*a4^3*b2^3*b3^2*b4^2 + 27*a1^4*a2^2*a3^3*a4^3*b1*b2^3*b3^2*b4^2 +
64*a1^3*a2^2*a3^3*a4^3*b1^2*b2^3*b3^2*b4^2 + 64*a1^2*a2^2*a3^3*a4^3*b1^3*b2^3*b3^2*b4^2 +
27*a1*a2^2*a3^3*a4^3*b1^4*b2^3*b3^2*b4^2 + 4*a2^2*a3^3*a4^3*b1^5*b2^3*b3^2*b4^2 +

2

a1^5*a2*a3^3*a4^3*b2^4*b3^2*b4^2 + 10*a1^4*a2*a3^3*a4^3*b1*b2^4*b3^2*b4^2 +
27*a1^3*a2*a3^3*a4^3*b1^2*b2^4*b3^2*b4^2 + 27*a1^2*a2*a3^3*a4^3*b1^3*b2^4*b3^2*b4^2 +
10*a1*a2*a3^3*a4^3*b1^4*b2^4*b3^2*b4^2 + a2*a3^3*a4^3*b1^5*b2^4*b3^2*b4^2 +
a1^4*a3^3*a4^3*b1*b2^5*b3^2*b4^2 + 4*a1^3*a3^3*a4^3*b1^2*b2^5*b3^2*b4^2 +
4*a1^2*a3^3*a4^3*b1^3*b2^5*b3^2*b4^2 + a1*a3^3*a4^3*b1^4*b2^5*b3^2*b4^2 +
a1^4*a2^5*a3^2*a4^3*b1*b3^3*b4^2 + 4*a1^3*a2^5*a3^2*a4^3*b1^2*b3^3*b4^2 +
4*a1^2*a2^5*a3^2*a4^3*b1^3*b3^3*b4^2 + a1*a2^5*a3^2*a4^3*b1^4*b3^3*b4^2 +
a1^5*a2^4*a3^2*a4^3*b2*b3^3*b4^2 + 10*a1^4*a2^4*a3^2*a4^3*b1*b2*b3^3*b4^2 +
27*a1^3*a2^4*a3^2*a4^3*b1^2*b2*b3^3*b4^2 + 27*a1^2*a2^4*a3^2*a4^3*b1^3*b2*b3^3*b4^2 +
10*a1*a2^4*a3^2*a4^3*b1^4*b2*b3^3*b4^2 + a2^4*a3^2*a4^3*b1^5*b2*b3^3*b4^2 +
4*a1^5*a2^3*a3^2*a4^3*b2^2*b3^3*b4^2 + 27*a1^4*a2^3*a3^2*a4^3*b1*b2^2*b3^3*b4^2 +
64*a1^3*a2^3*a3^2*a4^3*b1^2*b2^2*b3^3*b4^2 + 64*a1^2*a2^3*a3^2*a4^3*b1^3*b2^2*b3^3*b4^2 +
27*a1*a2^3*a3^2*a4^3*b1^4*b2^2*b3^3*b4^2 + 4*a2^3*a3^2*a4^3*b1^5*b2^2*b3^3*b4^2 +
4*a1^5*a2^2*a3^2*a4^3*b2^3*b3^3*b4^2 + 27*a1^4*a2^2*a3^2*a4^3*b1*b2^3*b3^3*b4^2 +
64*a1^3*a2^2*a3^2*a4^3*b1^2*b2^3*b3^3*b4^2 + 64*a1^2*a2^2*a3^2*a4^3*b1^3*b2^3*b3^3*b4^2 +
27*a1*a2^2*a3^2*a4^3*b1^4*b2^3*b3^3*b4^2 + 4*a2^2*a3^2*a4^3*b1^5*b2^3*b3^3*b4^2 +
a1^5*a2*a3^2*a4^3*b2^4*b3^3*b4^2 + 10*a1^4*a2*a3^2*a4^3*b1*b2^4*b3^3*b4^2 +
27*a1^3*a2*a3^2*a4^3*b1^2*b2^4*b3^3*b4^2 + 27*a1^2*a2*a3^2*a4^3*b1^3*b2^4*b3^3*b4^2 +
10*a1*a2*a3^2*a4^3*b1^4*b2^4*b3^3*b4^2 + a2*a3^2*a4^3*b1^5*b2^4*b3^3*b4^2 +
a1^4*a3^2*a4^3*b1*b2^5*b3^3*b4^2 + 4*a1^3*a3^2*a4^3*b1^2*b2^5*b3^3*b4^2 +
4*a1^2*a3^2*a4^3*b1^3*b2^5*b3^3*b4^2 + a1*a3^2*a4^3*b1^4*b2^5*b3^3*b4^2 +
a1^3*a2^5*a3*a4^3*b1^2*b3^4*b4^2 + a1^2*a2^5*a3*a4^3*b1^3*b3^4*b4^2 +
3*a1^4*a2^4*a3*a4^3*b1*b2*b3^4*b4^2 + 10*a1^3*a2^4*a3*a4^3*b1^2*b2*b3^4*b4^2 +
10*a1^2*a2^4*a3*a4^3*b1^3*b2*b3^4*b4^2 + 3*a1*a2^4*a3*a4^3*b1^4*b2*b3^4*b4^2 +
a1^5*a2^3*a3*a4^3*b2^2*b3^4*b4^2 + 10*a1^4*a2^3*a3*a4^3*b1*b2^2*b3^4*b4^2 +
27*a1^3*a2^3*a3*a4^3*b1^2*b2^2*b3^4*b4^2 + 27*a1^2*a2^3*a3*a4^3*b1^3*b2^2*b3^4*b4^2 +
10*a1*a2^3*a3*a4^3*b1^4*b2^2*b3^4*b4^2 + a2^3*a3*a4^3*b1^5*b2^2*b3^4*b4^2 +
a1^5*a2^2*a3*a4^3*b2^3*b3^4*b4^2 + 10*a1^4*a2^2*a3*a4^3*b1*b2^3*b3^4*b4^2 +
27*a1^3*a2^2*a3*a4^3*b1^2*b2^3*b3^4*b4^2 + 27*a1^2*a2^2*a3*a4^3*b1^3*b2^3*b3^4*b4^2 +
10*a1*a2^2*a3*a4^3*b1^4*b2^3*b3^4*b4^2 + a2^2*a3*a4^3*b1^5*b2^3*b3^4*b4^2 +
3*a1^4*a2*a3*a4^3*b1*b2^4*b3^4*b4^2 + 10*a1^3*a2*a3*a4^3*b1^2*b2^4*b3^4*b4^2 +
10*a1^2*a2*a3*a4^3*b1^3*b2^4*b3^4*b4^2 + 3*a1*a2*a3*a4^3*b1^4*b2^4*b3^4*b4^2 +
a1^3*a3*a4^3*b1^2*b2^5*b3^4*b4^2 + a1^2*a3*a4^3*b1^3*b2^5*b3^4*b4^2 +
a1^3*a2^4*a4^3*b1^2*b2*b3^5*b4^2 + a1^2*a2^4*a4^3*b1^3*b2*b3^5*b4^2 +
a1^4*a2^3*a4^3*b1*b2^2*b3^5*b4^2 + 4*a1^3*a2^3*a4^3*b1^2*b2^2*b3^5*b4^2 +
4*a1^2*a2^3*a4^3*b1^3*b2^2*b3^5*b4^2 + a1*a2^3*a4^3*b1^4*b2^2*b3^5*b4^2 +
a1^4*a2^2*a4^3*b1*b2^3*b3^5*b4^2 + 4*a1^3*a2^2*a4^3*b1^2*b2^3*b3^5*b4^2 +
4*a1^2*a2^2*a4^3*b1^3*b2^3*b3^5*b4^2 + a1*a2^2*a4^3*b1^4*b2^3*b3^5*b4^2 +
a1^3*a2*a4^3*b1^2*b2^4*b3^5*b4^2 + a1^2*a2*a4^3*b1^3*b2^4*b3^5*b4^2 +
a1^3*a2^4*a3^5*a4^2*b1^2*b2*b4^3 + a1^2*a2^4*a3^5*a4^2*b1^3*b2*b4^3 +
a1^4*a2^3*a3^5*a4^2*b1*b2^2*b4^3 + 4*a1^3*a2^3*a3^5*a4^2*b1^2*b2^2*b4^3 +
4*a1^2*a2^3*a3^5*a4^2*b1^3*b2^2*b4^3 + a1*a2^3*a3^5*a4^2*b1^4*b2^2*b4^3 +
a1^4*a2^2*a3^5*a4^2*b1*b2^3*b4^3 + 4*a1^3*a2^2*a3^5*a4^2*b1^2*b2^3*b4^3 +
4*a1^2*a2^2*a3^5*a4^2*b1^3*b2^3*b4^3 + a1*a2^2*a3^5*a4^2*b1^4*b2^3*b4^3 +
a1^3*a2*a3^5*a4^2*b1^2*b2^4*b4^3 + a1^2*a2*a3^5*a4^2*b1^3*b2^4*b4^3 +
a1^3*a2^5*a3^4*a4^2*b1^2*b3*b4^3 + a1^2*a2^5*a3^4*a4^2*b1^3*b3*b4^3 +
3*a1^4*a2^4*a3^4*a4^2*b1*b2*b3*b4^3 + 10*a1^3*a2^4*a3^4*a4^2*b1^2*b2*b3*b4^3 +
10*a1^2*a2^4*a3^4*a4^2*b1^3*b2*b3*b4^3 + 3*a1*a2^4*a3^4*a4^2*b1^4*b2*b3*b4^3 +
a1^5*a2^3*a3^4*a4^2*b2^2*b3*b4^3 + 10*a1^4*a2^3*a3^4*a4^2*b1*b2^2*b3*b4^3 +
27*a1^3*a2^3*a3^4*a4^2*b1^2*b2^2*b3*b4^3 + 27*a1^2*a2^3*a3^4*a4^2*b1^3*b2^2*b3*b4^3 +
10*a1*a2^3*a3^4*a4^2*b1^4*b2^2*b3*b4^3 + a2^3*a3^4*a4^2*b1^5*b2^2*b3*b4^3 +
a1^5*a2^2*a3^4*a4^2*b2^3*b3*b4^3 + 10*a1^4*a2^2*a3^4*a4^2*b1*b2^3*b3*b4^3 +
27*a1^3*a2^2*a3^4*a4^2*b1^2*b2^3*b3*b4^3 + 27*a1^2*a2^2*a3^4*a4^2*b1^3*b2^3*b3*b4^3 +
10*a1*a2^2*a3^4*a4^2*b1^4*b2^3*b3*b4^3 + a2^2*a3^4*a4^2*b1^5*b2^3*b3*b4^3 +
3*a1^4*a2*a3^4*a4^2*b1*b2^4*b3*b4^3 + 10*a1^3*a2*a3^4*a4^2*b1^2*b2^4*b3*b4^3 +
10*a1^2*a2*a3^4*a4^2*b1^3*b2^4*b3*b4^3 + 3*a1*a2*a3^4*a4^2*b1^4*b2^4*b3*b4^3 +
a1^3*a3^4*a4^2*b1^2*b2^5*b3*b4^3 + a1^2*a3^4*a4^2*b1^3*b2^5*b3*b4^3 +
a1^4*a2^5*a3^3*a4^2*b1*b3^2*b4^3 + 4*a1^3*a2^5*a3^3*a4^2*b1^2*b3^2*b4^3 +
4*a1^2*a2^5*a3^3*a4^2*b1^3*b3^2*b4^3 + a1*a2^5*a3^3*a4^2*b1^4*b3^2*b4^3 +
a1^5*a2^4*a3^3*a4^2*b2*b3^2*b4^3 + 10*a1^4*a2^4*a3^3*a4^2*b1*b2*b3^2*b4^3 +
27*a1^3*a2^4*a3^3*a4^2*b1^2*b2*b3^2*b4^3 + 27*a1^2*a2^4*a3^3*a4^2*b1^3*b2*b3^2*b4^3 +

3

10*a1*a2^4*a3^3*a4^2*b1^4*b2*b3^2*b4^3 + a2^4*a3^3*a4^2*b1^5*b2*b3^2*b4^3 +
4*a1^5*a2^3*a3^3*a4^2*b2^2*b3^2*b4^3 + 27*a1^4*a2^3*a3^3*a4^2*b1*b2^2*b3^2*b4^3 +
64*a1^3*a2^3*a3^3*a4^2*b1^2*b2^2*b3^2*b4^3 + 64*a1^2*a2^3*a3^3*a4^2*b1^3*b2^2*b3^2*b4^3
27*a1*a2^3*a3^3*a4^2*b1^4*b2^2*b3^2*b4^3 + 4*a2^3*a3^3*a4^2*b1^5*b2^2*b3^2*b4^3 +
4*a1^5*a2^2*a3^3*a4^2*b2^3*b3^2*b4^3 + 27*a1^4*a2^2*a3^3*a4^2*b1*b2^3*b3^2*b4^3 +
64*a1^3*a2^2*a3^3*a4^2*b1^2*b2^3*b3^2*b4^3 + 64*a1^2*a2^2*a3^3*a4^2*b1^3*b2^3*b3^2*b4^3
27*a1*a2^2*a3^3*a4^2*b1^4*b2^3*b3^2*b4^3 + 4*a2^2*a3^3*a4^2*b1^5*b2^3*b3^2*b4^3 +
a1^5*a2*a3^3*a4^2*b2^4*b3^2*b4^3 + 10*a1^4*a2*a3^3*a4^2*b1*b2^4*b3^2*b4^3 +
27*a1^3*a2*a3^3*a4^2*b1^2*b2^4*b3^2*b4^3 + 27*a1^2*a2*a3^3*a4^2*b1^3*b2^4*b3^2*b4^3 +
10*a1*a2*a3^3*a4^2*b1^4*b2^4*b3^2*b4^3 + a2*a3^3*a4^2*b1^5*b2^4*b3^2*b4^3 +
a1^4*a3^3*a4^2*b1*b2^5*b3^2*b4^3 + 4*a1^3*a3^3*a4^2*b1^2*b2^5*b3^2*b4^3 +
4*a1^2*a3^3*a4^2*b1^3*b2^5*b3^2*b4^3 + a1*a3^3*a4^2*b1^4*b2^5*b3^2*b4^3 +
a1^4*a2^5*a3^2*a4^2*b1*b3^3*b4^3 + 4*a1^3*a2^5*a3^2*a4^2*b1^2*b3^3*b4^3 +
4*a1^2*a2^5*a3^2*a4^2*b1^3*b3^3*b4^3 + a1*a2^5*a3^2*a4^2*b1^4*b3^3*b4^3 +
a1^5*a2^4*a3^2*a4^2*b2*b3^3*b4^3 + 10*a1^4*a2^4*a3^2*a4^2*b1*b2*b3^3*b4^3 +
27*a1^3*a2^4*a3^2*a4^2*b1^2*b2*b3^3*b4^3 + 27*a1^2*a2^4*a3^2*a4^2*b1^3*b2*b3^3*b4^3 +
10*a1*a2^4*a3^2*a4^2*b1^4*b2*b3^3*b4^3 + a2^4*a3^2*a4^2*b1^5*b2*b3^3*b4^3 +
4*a1^5*a2^3*a3^2*a4^2*b2^2*b3^3*b4^3 + 27*a1^4*a2^3*a3^2*a4^2*b1*b2^2*b3^3*b4^3 +
64*a1^3*a2^3*a3^2*a4^2*b1^2*b2^2*b3^3*b4^3 + 64*a1^2*a2^3*a3^2*a4^2*b1^3*b2^2*b3^3*b4^3
27*a1*a2^3*a3^2*a4^2*b1^4*b2^2*b3^3*b4^3 + 4*a2^3*a3^2*a4^2*b1^5*b2^2*b3^3*b4^3 +
4*a1^5*a2^2*a3^2*a4^2*b2^3*b3^3*b4^3 + 27*a1^4*a2^2*a3^2*a4^2*b1*b2^3*b3^3*b4^3 +
64*a1^3*a2^2*a3^2*a4^2*b1^2*b2^3*b3^3*b4^3 + 64*a1^2*a2^2*a3^2*a4^2*b1^3*b2^3*b3^3*b4^3
27*a1*a2^2*a3^2*a4^2*b1^4*b2^3*b3^3*b4^3 + 4*a2^2*a3^2*a4^2*b1^5*b2^3*b3^3*b4^3 +
a1^5*a2*a3^2*a4^2*b2^4*b3^3*b4^3 + 10*a1^4*a2*a3^2*a4^2*b1*b2^4*b3^3*b4^3 +
27*a1^3*a2*a3^2*a4^2*b1^2*b2^4*b3^3*b4^3 + 27*a1^2*a2*a3^2*a4^2*b1^3*b2^4*b3^3*b4^3 +
10*a1*a2*a3^2*a4^2*b1^4*b2^4*b3^3*b4^3 + a2*a3^2*a4^2*b1^5*b2^4*b3^3*b4^3 +
a1^4*a3^2*a4^2*b1*b2^5*b3^3*b4^3 + 4*a1^3*a3^2*a4^2*b1^2*b2^5*b3^3*b4^3 +
4*a1^2*a3^2*a4^2*b1^3*b2^5*b3^3*b4^3 + a1*a3^2*a4^2*b1^4*b2^5*b3^3*b4^3 +
a1^3*a2^5*a3*a4^2*b1^2*b3^4*b4^3 + a1^2*a2^5*a3*a4^2*b1^3*b3^4*b4^3 +
3*a1^4*a2^4*a3*a4^2*b1*b2*b3^4*b4^3 + 10*a1^3*a2^4*a3*a4^2*b1^2*b2*b3^4*b4^3 +
10*a1^2*a2^4*a3*a4^2*b1^3*b2*b3^4*b4^3 + 3*a1*a2^4*a3*a4^2*b1^4*b2*b3^4*b4^3 +
a1^5*a2^3*a3*a4^2*b2^2*b3^4*b4^3 + 10*a1^4*a2^3*a3*a4^2*b1*b2^2*b3^4*b4^3 +
27*a1^3*a2^3*a3*a4^2*b1^2*b2^2*b3^4*b4^3 + 27*a1^2*a2^3*a3*a4^2*b1^3*b2^2*b3^4*b4^3 +
10*a1*a2^3*a3*a4^2*b1^4*b2^2*b3^4*b4^3 + a2^3*a3*a4^2*b1^5*b2^2*b3^4*b4^3 +
a1^5*a2^2*a3*a4^2*b2^3*b3^4*b4^3 + 10*a1^4*a2^2*a3*a4^2*b1*b2^3*b3^4*b4^3 +
27*a1^3*a2^2*a3*a4^2*b1^2*b2^3*b3^4*b4^3 + 27*a1^2*a2^2*a3*a4^2*b1^3*b2^3*b3^4*b4^3 +
10*a1*a2^2*a3*a4^2*b1^4*b2^3*b3^4*b4^3 + a2^2*a3*a4^2*b1^5*b2^3*b3^4*b4^3 +
3*a1^4*a2*a3*a4^2*b1*b2^4*b3^4*b4^3 + 10*a1^3*a2*a3*a4^2*b1^2*b2^4*b3^4*b4^3 +
10*a1^2*a2*a3*a4^2*b1^3*b2^4*b3^4*b4^3 + 3*a1*a2*a3*a4^2*b1^4*b2^4*b3^4*b4^3 +
a1^3*a3*a4^2*b1^2*b2^5*b3^4*b4^3 + a1^2*a3*a4^2*b1^3*b2^5*b3^4*b4^3 +
a1^3*a2^4*a4^2*b1^2*b2*b3^5*b4^3 + a1^2*a2^4*a4^2*b1^3*b2*b3^5*b4^3 +
a1^4*a2^3*a4^2*b1*b2^2*b3^5*b4^3 + 4*a1^3*a2^3*a4^2*b1^2*b2^2*b3^5*b4^3 +
4*a1^2*a2^3*a4^2*b1^3*b2^2*b3^5*b4^3 + a1*a2^3*a4^2*b1^4*b2^2*b3^5*b4^3 +
a1^4*a2^2*a4^2*b1*b2^3*b3^5*b4^3 + 4*a1^3*a2^2*a4^2*b1^2*b2^3*b3^5*b4^3 +
4*a1^2*a2^2*a4^2*b1^3*b2^3*b3^5*b4^3 + a1*a2^2*a4^2*b1^4*b2^3*b3^5*b4^3 +
a1^3*a2*a4^2*b1^2*b2^4*b3^5*b4^3 + a1^2*a2*a4^2*b1^3*b2^4*b3^5*b4^3 +
a1^3*a2^3*a3^5*a4*b1^2*b2^2*b4^4 + a1^2*a2^3*a3^5*a4*b1^3*b2^2*b4^4 +
a1^3*a2^2*a3^5*a4*b1^2*b2^3*b4^4 + a1^2*a2^2*a3^5*a4*b1^3*b2^3*b4^4 +
a1^4*a2^4*a3^4*a4*b1*b2*b3*b4^4 + 3*a1^3*a2^4*a3^4*a4*b1^2*b2*b3*b4^4 +
3*a1^2*a2^4*a3^4*a4*b1^3*b2*b3*b4^4 + a1*a2^4*a3^4*a4*b1^4*b2*b3*b4^4 +
3*a1^4*a2^3*a3^4*a4*b1*b2^2*b3*b4^4 + 10*a1^3*a2^3*a3^4*a4*b1^2*b2^2*b3*b4^4 +
10*a1^2*a2^3*a3^4*a4*b1^3*b2^2*b3*b4^4 + 3*a1*a2^3*a3^4*a4*b1^4*b2^2*b3*b4^4 +
3*a1^4*a2^2*a3^4*a4*b1*b2^3*b3*b4^4 + 10*a1^3*a2^2*a3^4*a4*b1^2*b2^3*b3*b4^4 +
10*a1^2*a2^2*a3^4*a4*b1^3*b2^3*b3*b4^4 + 3*a1*a2^2*a3^4*a4*b1^4*b2^3*b3*b4^4 +
a1^4*a2*a3^4*a4*b1*b2^4*b3*b4^4 + 3*a1^3*a2*a3^4*a4*b1^2*b2^4*b3*b4^4 +
3*a1^2*a2*a3^4*a4*b1^3*b2^4*b3*b4^4 + a1*a2*a3^4*a4*b1^4*b2^4*b3*b4^4 +
a1^3*a2^5*a3^3*a4*b1^2*b3^2*b4^4 + a1^2*a2^5*a3^3*a4*b1^3*b3^2*b4^4 +
3*a1^4*a2^4*a3^3*a4*b1*b2*b3^2*b4^4 + 10*a1^3*a2^4*a3^3*a4*b1^2*b2*b3^2*b4^4 +
10*a1^2*a2^4*a3^3*a4*b1^3*b2*b3^2*b4^4 + 3*a1*a2^4*a3^3*a4*b1^4*b2*b3^2*b4^4 +
a1^5*a2^3*a3^3*a4*b2^2*b3^2*b4^4 + 10*a1^4*a2^3*a3^3*a4*b1*b2^2*b3^2*b4^4 +
27*a1^3*a2^3*a3^3*a4*b1^2*b2^2*b3^2*b4^4 + 27*a1^2*a2^3*a3^3*a4*b1^3*b2^2*b3^2*b4^4 +

4

+

+

+

+

10*a1*a2^3*a3^3*a4*b1^4*b2^2*b3^2*b4^4 + a2^3*a3^3*a4*b1^5*b2^2*b3^2*b4^4 +
a1^5*a2^2*a3^3*a4*b2^3*b3^2*b4^4 + 10*a1^4*a2^2*a3^3*a4*b1*b2^3*b3^2*b4^4 +
27*a1^3*a2^2*a3^3*a4*b1^2*b2^3*b3^2*b4^4 + 27*a1^2*a2^2*a3^3*a4*b1^3*b2^3*b3^2*b4^4 +
10*a1*a2^2*a3^3*a4*b1^4*b2^3*b3^2*b4^4 + a2^2*a3^3*a4*b1^5*b2^3*b3^2*b4^4 +
3*a1^4*a2*a3^3*a4*b1*b2^4*b3^2*b4^4 + 10*a1^3*a2*a3^3*a4*b1^2*b2^4*b3^2*b4^4 +
10*a1^2*a2*a3^3*a4*b1^3*b2^4*b3^2*b4^4 + 3*a1*a2*a3^3*a4*b1^4*b2^4*b3^2*b4^4 +
a1^3*a3^3*a4*b1^2*b2^5*b3^2*b4^4 + a1^2*a3^3*a4*b1^3*b2^5*b3^2*b4^4 +
a1^3*a2^5*a3^2*a4*b1^2*b3^3*b4^4 + a1^2*a2^5*a3^2*a4*b1^3*b3^3*b4^4 +
3*a1^4*a2^4*a3^2*a4*b1*b2*b3^3*b4^4 + 10*a1^3*a2^4*a3^2*a4*b1^2*b2*b3^3*b4^4 +
10*a1^2*a2^4*a3^2*a4*b1^3*b2*b3^3*b4^4 + 3*a1*a2^4*a3^2*a4*b1^4*b2*b3^3*b4^4 +
a1^5*a2^3*a3^2*a4*b2^2*b3^3*b4^4 + 10*a1^4*a2^3*a3^2*a4*b1*b2^2*b3^3*b4^4 +
27*a1^3*a2^3*a3^2*a4*b1^2*b2^2*b3^3*b4^4 + 27*a1^2*a2^3*a3^2*a4*b1^3*b2^2*b3^3*b4^4 +
10*a1*a2^3*a3^2*a4*b1^4*b2^2*b3^3*b4^4 + a2^3*a3^2*a4*b1^5*b2^2*b3^3*b4^4 +
a1^5*a2^2*a3^2*a4*b2^3*b3^3*b4^4 + 10*a1^4*a2^2*a3^2*a4*b1*b2^3*b3^3*b4^4 +
27*a1^3*a2^2*a3^2*a4*b1^2*b2^3*b3^3*b4^4 + 27*a1^2*a2^2*a3^2*a4*b1^3*b2^3*b3^3*b4^4 +
10*a1*a2^2*a3^2*a4*b1^4*b2^3*b3^3*b4^4 + a2^2*a3^2*a4*b1^5*b2^3*b3^3*b4^4 +
3*a1^4*a2*a3^2*a4*b1*b2^4*b3^3*b4^4 + 10*a1^3*a2*a3^2*a4*b1^2*b2^4*b3^3*b4^4 +
10*a1^2*a2*a3^2*a4*b1^3*b2^4*b3^3*b4^4 + 3*a1*a2*a3^2*a4*b1^4*b2^4*b3^3*b4^4 +
a1^3*a3^2*a4*b1^2*b2^5*b3^3*b4^4 + a1^2*a3^2*a4*b1^3*b2^5*b3^3*b4^4 +
a1^4*a2^4*a3*a4*b1*b2*b3^4*b4^4 + 3*a1^3*a2^4*a3*a4*b1^2*b2*b3^4*b4^4 +
3*a1^2*a2^4*a3*a4*b1^3*b2*b3^4*b4^4 + a1*a2^4*a3*a4*b1^4*b2*b3^4*b4^4 +
3*a1^4*a2^3*a3*a4*b1*b2^2*b3^4*b4^4 + 10*a1^3*a2^3*a3*a4*b1^2*b2^2*b3^4*b4^4 +
10*a1^2*a2^3*a3*a4*b1^3*b2^2*b3^4*b4^4 + 3*a1*a2^3*a3*a4*b1^4*b2^2*b3^4*b4^4 +
3*a1^4*a2^2*a3*a4*b1*b2^3*b3^4*b4^4 + 10*a1^3*a2^2*a3*a4*b1^2*b2^3*b3^4*b4^4 +
10*a1^2*a2^2*a3*a4*b1^3*b2^3*b3^4*b4^4 + 3*a1*a2^2*a3*a4*b1^4*b2^3*b3^4*b4^4 +
a1^4*a2*a3*a4*b1*b2^4*b3^4*b4^4 + 3*a1^3*a2*a3*a4*b1^2*b2^4*b3^4*b4^4 +
3*a1^2*a2*a3*a4*b1^3*b2^4*b3^4*b4^4 + a1*a2*a3*a4*b1^4*b2^4*b3^4*b4^4 +
a1^3*a2^3*a4*b1^2*b2^2*b3^5*b4^4 + a1^2*a2^3*a4*b1^3*b2^2*b3^5*b4^4 +
a1^3*a2^2*a4*b1^2*b2^3*b3^5*b4^4 + a1^2*a2^2*a4*b1^3*b2^3*b3^5*b4^4 +
a1^3*a2^3*a3^4*b1^2*b2^2*b3*b4^5 + a1^2*a2^3*a3^4*b1^3*b2^2*b3*b4^5 +
a1^3*a2^2*a3^4*b1^2*b2^3*b3*b4^5 + a1^2*a2^2*a3^4*b1^3*b2^3*b3*b4^5 +
a1^3*a2^4*a3^3*b1^2*b2*b3^2*b4^5 + a1^2*a2^4*a3^3*b1^3*b2*b3^2*b4^5 +
a1^4*a2^3*a3^3*b1*b2^2*b3^2*b4^5 + 4*a1^3*a2^3*a3^3*b1^2*b2^2*b3^2*b4^5 +
4*a1^2*a2^3*a3^3*b1^3*b2^2*b3^2*b4^5 + a1*a2^3*a3^3*b1^4*b2^2*b3^2*b4^5 +
a1^4*a2^2*a3^3*b1*b2^3*b3^2*b4^5 + 4*a1^3*a2^2*a3^3*b1^2*b2^3*b3^2*b4^5 +
4*a1^2*a2^2*a3^3*b1^3*b2^3*b3^2*b4^5 + a1*a2^2*a3^3*b1^4*b2^3*b3^2*b4^5 +
a1^3*a2*a3^3*b1^2*b2^4*b3^2*b4^5 + a1^2*a2*a3^3*b1^3*b2^4*b3^2*b4^5 +
a1^3*a2^4*a3^2*b1^2*b2*b3^3*b4^5 + a1^2*a2^4*a3^2*b1^3*b2*b3^3*b4^5 +
a1^4*a2^3*a3^2*b1*b2^2*b3^3*b4^5 + 4*a1^3*a2^3*a3^2*b1^2*b2^2*b3^3*b4^5 +
4*a1^2*a2^3*a3^2*b1^3*b2^2*b3^3*b4^5 + a1*a2^3*a3^2*b1^4*b2^2*b3^3*b4^5 +
a1^4*a2^2*a3^2*b1*b2^3*b3^3*b4^5 + 4*a1^3*a2^2*a3^2*b1^2*b2^3*b3^3*b4^5 +
4*a1^2*a2^2*a3^2*b1^3*b2^3*b3^3*b4^5 + a1*a2^2*a3^2*b1^4*b2^3*b3^3*b4^5 +
a1^3*a2*a3^2*b1^2*b2^4*b3^3*b4^5 + a1^2*a2*a3^2*b1^3*b2^4*b3^3*b4^5 +
a1^3*a2^3*a3*b1^2*b2^2*b3^4*b4^5 + a1^2*a2^3*a3*b1^3*b2^2*b3^4*b4^5 +
a1^3*a2^2*a3*b1^2*b2^3*b3^4*b4^5 + a1^2*a2^2*a3*b1^3*b2^3*b3^4*b4^5
g11 . f a c t o r ( )
#This i s an i n i t i a l f a c t o r i n g t o t r y and make t h i n g s e a s i e r .
(a1^2*a2^2*a3^3*a4^4*b1^2*b2^2*b3 + a1^2*a2^3*a3^2*a4^4*b1^2*b2*b3^2 +
a1^3*a2^2*a3^2*a4^4*b1*b2^2*b3^2 + a1^2*a2^2*a3^2*a4^4*b1^2*b2^2*b3^2 +
a1*a2^2*a3^2*a4^4*b1^3*b2^2*b3^2 + a1^2*a2*a3^2*a4^4*b1^2*b2^3*b3^2 +
a1^2*a2^2*a3*a4^4*b1^2*b2^2*b3^3 + a1^2*a2^2*a3^4*a4^3*b1^2*b2^2*b4 +
a1^3*a2^3*a3^3*a4^3*b1*b2*b3*b4 + 2*a1^2*a2^3*a3^3*a4^3*b1^2*b2*b3*b4 +
a1*a2^3*a3^3*a4^3*b1^3*b2*b3*b4 + 2*a1^3*a2^2*a3^3*a4^3*b1*b2^2*b3*b4 +
3*a1^2*a2^2*a3^3*a4^3*b1^2*b2^2*b3*b4 + 2*a1*a2^2*a3^3*a4^3*b1^3*b2^2*b3*b4 +
a1^3*a2*a3^3*a4^3*b1*b2^3*b3*b4 + 2*a1^2*a2*a3^3*a4^3*b1^2*b2^3*b3*b4 +
a1*a2*a3^3*a4^3*b1^3*b2^3*b3*b4 + a1^2*a2^4*a3^2*a4^3*b1^2*b3^2*b4 +
2*a1^3*a2^3*a3^2*a4^3*b1*b2*b3^2*b4 + 3*a1^2*a2^3*a3^2*a4^3*b1^2*b2*b3^2*b4 +
2*a1*a2^3*a3^2*a4^3*b1^3*b2*b3^2*b4 + a1^4*a2^2*a3^2*a4^3*b2^2*b3^2*b4 +
3*a1^3*a2^2*a3^2*a4^3*b1*b2^2*b3^2*b4 + 7*a1^2*a2^2*a3^2*a4^3*b1^2*b2^2*b3^2*b4 +

5

3*a1*a2^2*a3^2*a4^3*b1^3*b2^2*b3^2*b4 + a2^2*a3^2*a4^3*b1^4*b2^2*b3^2*b4 +
2*a1^3*a2*a3^2*a4^3*b1*b2^3*b3^2*b4 + 3*a1^2*a2*a3^2*a4^3*b1^2*b2^3*b3^2*b4 +
2*a1*a2*a3^2*a4^3*b1^3*b2^3*b3^2*b4 + a1^2*a3^2*a4^3*b1^2*b2^4*b3^2*b4 +
a1^3*a2^3*a3*a4^3*b1*b2*b3^3*b4 + 2*a1^2*a2^3*a3*a4^3*b1^2*b2*b3^3*b4 +
a1*a2^3*a3*a4^3*b1^3*b2*b3^3*b4 + 2*a1^3*a2^2*a3*a4^3*b1*b2^2*b3^3*b4 +
3*a1^2*a2^2*a3*a4^3*b1^2*b2^2*b3^3*b4 + 2*a1*a2^2*a3*a4^3*b1^3*b2^2*b3^3*b4 +
a1^3*a2*a3*a4^3*b1*b2^3*b3^3*b4 + 2*a1^2*a2*a3*a4^3*b1^2*b2^3*b3^3*b4 +
a1*a2*a3*a4^3*b1^3*b2^3*b3^3*b4 + a1^2*a2^2*a4^3*b1^2*b2^2*b3^4*b4 +
a1^2*a2^3*a3^4*a4^2*b1^2*b2*b4^2 + a1^3*a2^2*a3^4*a4^2*b1*b2^2*b4^2 +
a1^2*a2^2*a3^4*a4^2*b1^2*b2^2*b4^2 + a1*a2^2*a3^4*a4^2*b1^3*b2^2*b4^2 +
a1^2*a2*a3^4*a4^2*b1^2*b2^3*b4^2 + a1^2*a2^4*a3^3*a4^2*b1^2*b3*b4^2 +
2*a1^3*a2^3*a3^3*a4^2*b1*b2*b3*b4^2 + 3*a1^2*a2^3*a3^3*a4^2*b1^2*b2*b3*b4^2 +
2*a1*a2^3*a3^3*a4^2*b1^3*b2*b3*b4^2 + a1^4*a2^2*a3^3*a4^2*b2^2*b3*b4^2 +
3*a1^3*a2^2*a3^3*a4^2*b1*b2^2*b3*b4^2 + 7*a1^2*a2^2*a3^3*a4^2*b1^2*b2^2*b3*b4^2 +
3*a1*a2^2*a3^3*a4^2*b1^3*b2^2*b3*b4^2 + a2^2*a3^3*a4^2*b1^4*b2^2*b3*b4^2 +
2*a1^3*a2*a3^3*a4^2*b1*b2^3*b3*b4^2 + 3*a1^2*a2*a3^3*a4^2*b1^2*b2^3*b3*b4^2 +
2*a1*a2*a3^3*a4^2*b1^3*b2^3*b3*b4^2 + a1^2*a3^3*a4^2*b1^2*b2^4*b3*b4^2 +
a1^3*a2^4*a3^2*a4^2*b1*b3^2*b4^2 + a1^2*a2^4*a3^2*a4^2*b1^2*b3^2*b4^2 +
a1*a2^4*a3^2*a4^2*b1^3*b3^2*b4^2 + a1^4*a2^3*a3^2*a4^2*b2*b3^2*b4^2 +
3*a1^3*a2^3*a3^2*a4^2*b1*b2*b3^2*b4^2 + 7*a1^2*a2^3*a3^2*a4^2*b1^2*b2*b3^2*b4^2 +
3*a1*a2^3*a3^2*a4^2*b1^3*b2*b3^2*b4^2 + a2^3*a3^2*a4^2*b1^4*b2*b3^2*b4^2 +
a1^4*a2^2*a3^2*a4^2*b2^2*b3^2*b4^2 + 7*a1^3*a2^2*a3^2*a4^2*b1*b2^2*b3^2*b4^2 +
9*a1^2*a2^2*a3^2*a4^2*b1^2*b2^2*b3^2*b4^2 + 7*a1*a2^2*a3^2*a4^2*b1^3*b2^2*b3^2*b4^2 +
a2^2*a3^2*a4^2*b1^4*b2^2*b3^2*b4^2 + a1^4*a2*a3^2*a4^2*b2^3*b3^2*b4^2 +
3*a1^3*a2*a3^2*a4^2*b1*b2^3*b3^2*b4^2 + 7*a1^2*a2*a3^2*a4^2*b1^2*b2^3*b3^2*b4^2 +
3*a1*a2*a3^2*a4^2*b1^3*b2^3*b3^2*b4^2 + a2*a3^2*a4^2*b1^4*b2^3*b3^2*b4^2 +
a1^3*a3^2*a4^2*b1*b2^4*b3^2*b4^2 + a1^2*a3^2*a4^2*b1^2*b2^4*b3^2*b4^2 +
a1*a3^2*a4^2*b1^3*b2^4*b3^2*b4^2 + a1^2*a2^4*a3*a4^2*b1^2*b3^3*b4^2 +
2*a1^3*a2^3*a3*a4^2*b1*b2*b3^3*b4^2 + 3*a1^2*a2^3*a3*a4^2*b1^2*b2*b3^3*b4^2 +
2*a1*a2^3*a3*a4^2*b1^3*b2*b3^3*b4^2 + a1^4*a2^2*a3*a4^2*b2^2*b3^3*b4^2 +
3*a1^3*a2^2*a3*a4^2*b1*b2^2*b3^3*b4^2 + 7*a1^2*a2^2*a3*a4^2*b1^2*b2^2*b3^3*b4^2 +
3*a1*a2^2*a3*a4^2*b1^3*b2^2*b3^3*b4^2 + a2^2*a3*a4^2*b1^4*b2^2*b3^3*b4^2 +
2*a1^3*a2*a3*a4^2*b1*b2^3*b3^3*b4^2 + 3*a1^2*a2*a3*a4^2*b1^2*b2^3*b3^3*b4^2 +
2*a1*a2*a3*a4^2*b1^3*b2^3*b3^3*b4^2 + a1^2*a3*a4^2*b1^2*b2^4*b3^3*b4^2 +
a1^2*a2^3*a4^2*b1^2*b2*b3^4*b4^2 + a1^3*a2^2*a4^2*b1*b2^2*b3^4*b4^2 +
a1^2*a2^2*a4^2*b1^2*b2^2*b3^4*b4^2 + a1*a2^2*a4^2*b1^3*b2^2*b3^4*b4^2 +
a1^2*a2*a4^2*b1^2*b2^3*b3^4*b4^2 + a1^2*a2^2*a3^4*a4*b1^2*b2^2*b4^3 +
a1^3*a2^3*a3^3*a4*b1*b2*b3*b4^3 + 2*a1^2*a2^3*a3^3*a4*b1^2*b2*b3*b4^3 +
a1*a2^3*a3^3*a4*b1^3*b2*b3*b4^3 + 2*a1^3*a2^2*a3^3*a4*b1*b2^2*b3*b4^3 +
3*a1^2*a2^2*a3^3*a4*b1^2*b2^2*b3*b4^3 + 2*a1*a2^2*a3^3*a4*b1^3*b2^2*b3*b4^3 +
a1^3*a2*a3^3*a4*b1*b2^3*b3*b4^3 + 2*a1^2*a2*a3^3*a4*b1^2*b2^3*b3*b4^3 +
a1*a2*a3^3*a4*b1^3*b2^3*b3*b4^3 + a1^2*a2^4*a3^2*a4*b1^2*b3^2*b4^3 +
2*a1^3*a2^3*a3^2*a4*b1*b2*b3^2*b4^3 + 3*a1^2*a2^3*a3^2*a4*b1^2*b2*b3^2*b4^3 +
2*a1*a2^3*a3^2*a4*b1^3*b2*b3^2*b4^3 + a1^4*a2^2*a3^2*a4*b2^2*b3^2*b4^3 +
3*a1^3*a2^2*a3^2*a4*b1*b2^2*b3^2*b4^3 + 7*a1^2*a2^2*a3^2*a4*b1^2*b2^2*b3^2*b4^3 +
3*a1*a2^2*a3^2*a4*b1^3*b2^2*b3^2*b4^3 + a2^2*a3^2*a4*b1^4*b2^2*b3^2*b4^3 +
2*a1^3*a2*a3^2*a4*b1*b2^3*b3^2*b4^3 + 3*a1^2*a2*a3^2*a4*b1^2*b2^3*b3^2*b4^3 +
2*a1*a2*a3^2*a4*b1^3*b2^3*b3^2*b4^3 + a1^2*a3^2*a4*b1^2*b2^4*b3^2*b4^3 +
a1^3*a2^3*a3*a4*b1*b2*b3^3*b4^3 + 2*a1^2*a2^3*a3*a4*b1^2*b2*b3^3*b4^3 +
a1*a2^3*a3*a4*b1^3*b2*b3^3*b4^3 + 2*a1^3*a2^2*a3*a4*b1*b2^2*b3^3*b4^3 +
3*a1^2*a2^2*a3*a4*b1^2*b2^2*b3^3*b4^3 + 2*a1*a2^2*a3*a4*b1^3*b2^2*b3^3*b4^3 +
a1^3*a2*a3*a4*b1*b2^3*b3^3*b4^3 + 2*a1^2*a2*a3*a4*b1^2*b2^3*b3^3*b4^3 +
a1*a2*a3*a4*b1^3*b2^3*b3^3*b4^3 + a1^2*a2^2*a4*b1^2*b2^2*b3^4*b4^3 +
a1^2*a2^2*a3^3*b1^2*b2^2*b3*b4^4 + a1^2*a2^3*a3^2*b1^2*b2*b3^2*b4^4 +
a1^3*a2^2*a3^2*b1*b2^2*b3^2*b4^4 + a1^2*a2^2*a3^2*b1^2*b2^2*b3^2*b4^4 +
a1*a2^2*a3^2*b1^3*b2^2*b3^2*b4^4 + a1^2*a2*a3^2*b1^2*b2^3*b3^2*b4^4 +
a1^2*a2^2*a3*b1^2*b2^2*b3^3*b4^4)*(a1 + b1)*(a2 + b2)*(a3 + b3)*(a4 + b4)
t =(a1 + b1 ) * ( a2 + b2 ) * ( a3 + b3 ) * ( a4 + b4 )
t

6

f 1 1=g11 . f a c t o r ( ) / t
f11
#This s e p e r a t e s t h e f a c t o r i z a t i o n above i n t o two p a r t s .
(a1 + b1)*(a2 + b2)*(a3 + b3)*(a4 + b4)
a1^2*a2^2*a3^3*a4^4*b1^2*b2^2*b3 + a1^2*a2^3*a3^2*a4^4*b1^2*b2*b3^2 +
a1^3*a2^2*a3^2*a4^4*b1*b2^2*b3^2 + a1^2*a2^2*a3^2*a4^4*b1^2*b2^2*b3^2 +
a1*a2^2*a3^2*a4^4*b1^3*b2^2*b3^2 + a1^2*a2*a3^2*a4^4*b1^2*b2^3*b3^2 +
a1^2*a2^2*a3*a4^4*b1^2*b2^2*b3^3 + a1^2*a2^2*a3^4*a4^3*b1^2*b2^2*b4 +
a1^3*a2^3*a3^3*a4^3*b1*b2*b3*b4 + 2*a1^2*a2^3*a3^3*a4^3*b1^2*b2*b3*b4 +
a1*a2^3*a3^3*a4^3*b1^3*b2*b3*b4 + 2*a1^3*a2^2*a3^3*a4^3*b1*b2^2*b3*b4 +
3*a1^2*a2^2*a3^3*a4^3*b1^2*b2^2*b3*b4 + 2*a1*a2^2*a3^3*a4^3*b1^3*b2^2*b3*b4 +
a1^3*a2*a3^3*a4^3*b1*b2^3*b3*b4 + 2*a1^2*a2*a3^3*a4^3*b1^2*b2^3*b3*b4 +
a1*a2*a3^3*a4^3*b1^3*b2^3*b3*b4 + a1^2*a2^4*a3^2*a4^3*b1^2*b3^2*b4 +
2*a1^3*a2^3*a3^2*a4^3*b1*b2*b3^2*b4 + 3*a1^2*a2^3*a3^2*a4^3*b1^2*b2*b3^2*b4 +
2*a1*a2^3*a3^2*a4^3*b1^3*b2*b3^2*b4 + a1^4*a2^2*a3^2*a4^3*b2^2*b3^2*b4 +
3*a1^3*a2^2*a3^2*a4^3*b1*b2^2*b3^2*b4 + 7*a1^2*a2^2*a3^2*a4^3*b1^2*b2^2*b3^2*b4 +
3*a1*a2^2*a3^2*a4^3*b1^3*b2^2*b3^2*b4 + a2^2*a3^2*a4^3*b1^4*b2^2*b3^2*b4 +
2*a1^3*a2*a3^2*a4^3*b1*b2^3*b3^2*b4 + 3*a1^2*a2*a3^2*a4^3*b1^2*b2^3*b3^2*b4 +
2*a1*a2*a3^2*a4^3*b1^3*b2^3*b3^2*b4 + a1^2*a3^2*a4^3*b1^2*b2^4*b3^2*b4 +
a1^3*a2^3*a3*a4^3*b1*b2*b3^3*b4 + 2*a1^2*a2^3*a3*a4^3*b1^2*b2*b3^3*b4 +
a1*a2^3*a3*a4^3*b1^3*b2*b3^3*b4 + 2*a1^3*a2^2*a3*a4^3*b1*b2^2*b3^3*b4 +
3*a1^2*a2^2*a3*a4^3*b1^2*b2^2*b3^3*b4 + 2*a1*a2^2*a3*a4^3*b1^3*b2^2*b3^3*b4 +
a1^3*a2*a3*a4^3*b1*b2^3*b3^3*b4 + 2*a1^2*a2*a3*a4^3*b1^2*b2^3*b3^3*b4 +
a1*a2*a3*a4^3*b1^3*b2^3*b3^3*b4 + a1^2*a2^2*a4^3*b1^2*b2^2*b3^4*b4 +
a1^2*a2^3*a3^4*a4^2*b1^2*b2*b4^2 + a1^3*a2^2*a3^4*a4^2*b1*b2^2*b4^2 +
a1^2*a2^2*a3^4*a4^2*b1^2*b2^2*b4^2 + a1*a2^2*a3^4*a4^2*b1^3*b2^2*b4^2 +
a1^2*a2*a3^4*a4^2*b1^2*b2^3*b4^2 + a1^2*a2^4*a3^3*a4^2*b1^2*b3*b4^2 +
2*a1^3*a2^3*a3^3*a4^2*b1*b2*b3*b4^2 + 3*a1^2*a2^3*a3^3*a4^2*b1^2*b2*b3*b4^2 +
2*a1*a2^3*a3^3*a4^2*b1^3*b2*b3*b4^2 + a1^4*a2^2*a3^3*a4^2*b2^2*b3*b4^2 +
3*a1^3*a2^2*a3^3*a4^2*b1*b2^2*b3*b4^2 + 7*a1^2*a2^2*a3^3*a4^2*b1^2*b2^2*b3*b4^2 +
3*a1*a2^2*a3^3*a4^2*b1^3*b2^2*b3*b4^2 + a2^2*a3^3*a4^2*b1^4*b2^2*b3*b4^2 +
2*a1^3*a2*a3^3*a4^2*b1*b2^3*b3*b4^2 + 3*a1^2*a2*a3^3*a4^2*b1^2*b2^3*b3*b4^2 +
2*a1*a2*a3^3*a4^2*b1^3*b2^3*b3*b4^2 + a1^2*a3^3*a4^2*b1^2*b2^4*b3*b4^2 +
a1^3*a2^4*a3^2*a4^2*b1*b3^2*b4^2 + a1^2*a2^4*a3^2*a4^2*b1^2*b3^2*b4^2 +
a1*a2^4*a3^2*a4^2*b1^3*b3^2*b4^2 + a1^4*a2^3*a3^2*a4^2*b2*b3^2*b4^2 +
3*a1^3*a2^3*a3^2*a4^2*b1*b2*b3^2*b4^2 + 7*a1^2*a2^3*a3^2*a4^2*b1^2*b2*b3^2*b4^2 +
3*a1*a2^3*a3^2*a4^2*b1^3*b2*b3^2*b4^2 + a2^3*a3^2*a4^2*b1^4*b2*b3^2*b4^2 +
a1^4*a2^2*a3^2*a4^2*b2^2*b3^2*b4^2 + 7*a1^3*a2^2*a3^2*a4^2*b1*b2^2*b3^2*b4^2 +
9*a1^2*a2^2*a3^2*a4^2*b1^2*b2^2*b3^2*b4^2 + 7*a1*a2^2*a3^2*a4^2*b1^3*b2^2*b3^2*b4^2 +
a2^2*a3^2*a4^2*b1^4*b2^2*b3^2*b4^2 + a1^4*a2*a3^2*a4^2*b2^3*b3^2*b4^2 +
3*a1^3*a2*a3^2*a4^2*b1*b2^3*b3^2*b4^2 + 7*a1^2*a2*a3^2*a4^2*b1^2*b2^3*b3^2*b4^2 +
3*a1*a2*a3^2*a4^2*b1^3*b2^3*b3^2*b4^2 + a2*a3^2*a4^2*b1^4*b2^3*b3^2*b4^2 +
a1^3*a3^2*a4^2*b1*b2^4*b3^2*b4^2 + a1^2*a3^2*a4^2*b1^2*b2^4*b3^2*b4^2 +
a1*a3^2*a4^2*b1^3*b2^4*b3^2*b4^2 + a1^2*a2^4*a3*a4^2*b1^2*b3^3*b4^2 +
2*a1^3*a2^3*a3*a4^2*b1*b2*b3^3*b4^2 + 3*a1^2*a2^3*a3*a4^2*b1^2*b2*b3^3*b4^2 +
2*a1*a2^3*a3*a4^2*b1^3*b2*b3^3*b4^2 + a1^4*a2^2*a3*a4^2*b2^2*b3^3*b4^2 +
3*a1^3*a2^2*a3*a4^2*b1*b2^2*b3^3*b4^2 + 7*a1^2*a2^2*a3*a4^2*b1^2*b2^2*b3^3*b4^2 +
3*a1*a2^2*a3*a4^2*b1^3*b2^2*b3^3*b4^2 + a2^2*a3*a4^2*b1^4*b2^2*b3^3*b4^2 +
2*a1^3*a2*a3*a4^2*b1*b2^3*b3^3*b4^2 + 3*a1^2*a2*a3*a4^2*b1^2*b2^3*b3^3*b4^2 +
2*a1*a2*a3*a4^2*b1^3*b2^3*b3^3*b4^2 + a1^2*a3*a4^2*b1^2*b2^4*b3^3*b4^2 +
a1^2*a2^3*a4^2*b1^2*b2*b3^4*b4^2 + a1^3*a2^2*a4^2*b1*b2^2*b3^4*b4^2 +
a1^2*a2^2*a4^2*b1^2*b2^2*b3^4*b4^2 + a1*a2^2*a4^2*b1^3*b2^2*b3^4*b4^2 +
a1^2*a2*a4^2*b1^2*b2^3*b3^4*b4^2 + a1^2*a2^2*a3^4*a4*b1^2*b2^2*b4^3 +
a1^3*a2^3*a3^3*a4*b1*b2*b3*b4^3 + 2*a1^2*a2^3*a3^3*a4*b1^2*b2*b3*b4^3 +
a1*a2^3*a3^3*a4*b1^3*b2*b3*b4^3 + 2*a1^3*a2^2*a3^3*a4*b1*b2^2*b3*b4^3 +
3*a1^2*a2^2*a3^3*a4*b1^2*b2^2*b3*b4^3 + 2*a1*a2^2*a3^3*a4*b1^3*b2^2*b3*b4^3 +
a1^3*a2*a3^3*a4*b1*b2^3*b3*b4^3 + 2*a1^2*a2*a3^3*a4*b1^2*b2^3*b3*b4^3 +
a1*a2*a3^3*a4*b1^3*b2^3*b3*b4^3 + a1^2*a2^4*a3^2*a4*b1^2*b3^2*b4^3 +
2*a1^3*a2^3*a3^2*a4*b1*b2*b3^2*b4^3 + 3*a1^2*a2^3*a3^2*a4*b1^2*b2*b3^2*b4^3 +
2*a1*a2^3*a3^2*a4*b1^3*b2*b3^2*b4^3 + a1^4*a2^2*a3^2*a4*b2^2*b3^2*b4^3 +
3*a1^3*a2^2*a3^2*a4*b1*b2^2*b3^2*b4^3 + 7*a1^2*a2^2*a3^2*a4*b1^2*b2^2*b3^2*b4^3 +

7

3*a1*a2^2*a3^2*a4*b1^3*b2^2*b3^2*b4^3 + a2^2*a3^2*a4*b1^4*b2^2*b3^2*b4^3 +
2*a1^3*a2*a3^2*a4*b1*b2^3*b3^2*b4^3 + 3*a1^2*a2*a3^2*a4*b1^2*b2^3*b3^2*b4^3 +
2*a1*a2*a3^2*a4*b1^3*b2^3*b3^2*b4^3 + a1^2*a3^2*a4*b1^2*b2^4*b3^2*b4^3 +
a1^3*a2^3*a3*a4*b1*b2*b3^3*b4^3 + 2*a1^2*a2^3*a3*a4*b1^2*b2*b3^3*b4^3 +
a1*a2^3*a3*a4*b1^3*b2*b3^3*b4^3 + 2*a1^3*a2^2*a3*a4*b1*b2^2*b3^3*b4^3 +
3*a1^2*a2^2*a3*a4*b1^2*b2^2*b3^3*b4^3 + 2*a1*a2^2*a3*a4*b1^3*b2^2*b3^3*b4^3 +
a1^3*a2*a3*a4*b1*b2^3*b3^3*b4^3 + 2*a1^2*a2*a3*a4*b1^2*b2^3*b3^3*b4^3 +
a1*a2*a3*a4*b1^3*b2^3*b3^3*b4^3 + a1^2*a2^2*a4*b1^2*b2^2*b3^4*b4^3 +
a1^2*a2^2*a3^3*b1^2*b2^2*b3*b4^4 + a1^2*a2^3*a3^2*b1^2*b2*b3^2*b4^4 +
a1^3*a2^2*a3^2*b1*b2^2*b3^2*b4^4 + a1^2*a2^2*a3^2*b1^2*b2^2*b3^2*b4^4 +
a1*a2^2*a3^2*b1^3*b2^2*b3^2*b4^4 + a1^2*a2*a3^2*b1^2*b2^3*b3^2*b4^4 +
a1^2*a2^2*a3*b1^2*b2^2*b3^3*b4^4
s=p1*p2*p3*p4
s
#This i s our s u b s t i t u t i o n f o r t .
p1*p2*p3*p4
c2=sum ( f 1 1 . f i n d ( 2 *w0) )
c2
#This f i n d s a l l our terms i n f 1 1
t o g e t h e r . The gr oups we form
v a r i a b l e both t o t h e se c o n d
2*a1^2*a2^3*a3^3*a4^3*b1^2*b2*b3*b4 +
2*a1*a2^2*a3^3*a4^3*b1^3*b2^2*b3*b4 +
2*a1^3*a2^3*a3^2*a4^3*b1*b2*b3^2*b4 +
2*a1^3*a2*a3^2*a4^3*b1*b2^3*b3^2*b4 +
2*a1^2*a2^3*a3*a4^3*b1^2*b2*b3^3*b4 +
2*a1*a2^2*a3*a4^3*b1^3*b2^2*b3^3*b4 +
2*a1^3*a2^3*a3^3*a4^2*b1*b2*b3*b4^2 +
2*a1^3*a2*a3^3*a4^2*b1*b2^3*b3*b4^2 +
2*a1^3*a2^3*a3*a4^2*b1*b2*b3^3*b4^2 +
2*a1^3*a2*a3*a4^2*b1*b2^3*b3^3*b4^2 +
2*a1^2*a2^3*a3^3*a4*b1^2*b2*b3*b4^3 +
2*a1*a2^2*a3^3*a4*b1^3*b2^2*b3*b4^3 +
2*a1^3*a2^3*a3^2*a4*b1*b2*b3^2*b4^3 +
2*a1^3*a2*a3^2*a4*b1*b2^3*b3^2*b4^3 +
2*a1^2*a2^3*a3*a4*b1^2*b2*b3^3*b4^3 +
2*a1*a2^2*a3*a4*b1^3*b2^2*b3^3*b4^3 +

t h a t have a 2 i n them and then adds them \
w i l l have an a v a r i a b l e and i t s c o r r e s p o n d i n g b\
power .
2*a1^3*a2^2*a3^3*a4^3*b1*b2^2*b3*b4 +
2*a1^2*a2*a3^3*a4^3*b1^2*b2^3*b3*b4 +
2*a1*a2^3*a3^2*a4^3*b1^3*b2*b3^2*b4 +
2*a1*a2*a3^2*a4^3*b1^3*b2^3*b3^2*b4 +
2*a1^3*a2^2*a3*a4^3*b1*b2^2*b3^3*b4 +
2*a1^2*a2*a3*a4^3*b1^2*b2^3*b3^3*b4 +
2*a1*a2^3*a3^3*a4^2*b1^3*b2*b3*b4^2 +
2*a1*a2*a3^3*a4^2*b1^3*b2^3*b3*b4^2 +
2*a1*a2^3*a3*a4^2*b1^3*b2*b3^3*b4^2 +
2*a1*a2*a3*a4^2*b1^3*b2^3*b3^3*b4^2 +
2*a1^3*a2^2*a3^3*a4*b1*b2^2*b3*b4^3 +
2*a1^2*a2*a3^3*a4*b1^2*b2^3*b3*b4^3 +
2*a1*a2^3*a3^2*a4*b1^3*b2*b3^2*b4^3 +
2*a1*a2*a3^2*a4*b1^3*b2^3*b3^2*b4^3 +
2*a1^3*a2^2*a3*a4*b1*b2^2*b3^3*b4^3 +
2*a1^2*a2*a3*a4*b1^2*b2^3*b3^3*b4^3

c211=sum ( c2 . f i n d ( a1 ^2* b1 ^2*w0) )
c211
c211 . f a c t o r ( )
s 2 1 1 = -2*( p2^2+2*q2 ) * ( p3^2+2*q3 ) * ( p4^2+2*q4 ) * q1 ^2* q2 * q3 * q4
s211
#This i s t h e sum o f t h e terms with a1 ^2 and b1 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
2*a1^2*a2^3*a3^3*a4^3*b1^2*b2*b3*b4 + 2*a1^2*a2*a3^3*a4^3*b1^2*b2^3*b3*b4 +
2*a1^2*a2^3*a3*a4^3*b1^2*b2*b3^3*b4 + 2*a1^2*a2*a3*a4^3*b1^2*b2^3*b3^3*b4 +
2*a1^2*a2^3*a3^3*a4*b1^2*b2*b3*b4^3 + 2*a1^2*a2*a3^3*a4*b1^2*b2^3*b3*b4^3 +
2*a1^2*a2^3*a3*a4*b1^2*b2*b3^3*b4^3 + 2*a1^2*a2*a3*a4*b1^2*b2^3*b3^3*b4^3
2*(a2^2 + b2^2)*(a3^2 + b3^2)*(a4^2 + b4^2)*a1^2*a2*a3*a4*b1^2*b2*b3*b4
-2*(p2^2 + 2*q2)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1^2*q2*q3*q4
c222=sum ( c2 . f i n d ( a2 ^2* b2 ^2*w0) )
c222
c222 . f a c t o r ( )
s 2 2 2 = -2*( p1^2+2*q1 ) * ( p3^2+2*q3 ) * ( p4^2+2*q4 ) * q1 * q2 ^2* q3 * q4
s222
#This i s t h e sum o f t h e terms with a2 ^2 and b2 ^2 , i t s f a c t o r i z a t i o n , and i t s \

8

substitution .
2*a1^3*a2^2*a3^3*a4^3*b1*b2^2*b3*b4 + 2*a1*a2^2*a3^3*a4^3*b1^3*b2^2*b3*b4 +
2*a1^3*a2^2*a3*a4^3*b1*b2^2*b3^3*b4 + 2*a1*a2^2*a3*a4^3*b1^3*b2^2*b3^3*b4 +
2*a1^3*a2^2*a3^3*a4*b1*b2^2*b3*b4^3 + 2*a1*a2^2*a3^3*a4*b1^3*b2^2*b3*b4^3 +
2*a1^3*a2^2*a3*a4*b1*b2^2*b3^3*b4^3 + 2*a1*a2^2*a3*a4*b1^3*b2^2*b3^3*b4^3
2*(a1^2 + b1^2)*(a3^2 + b3^2)*(a4^2 + b4^2)*a1*a2^2*a3*a4*b1*b2^2*b3*b4
-2*(p1^2 + 2*q1)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1*q2^2*q3*q4
c233=sum ( c2 . f i n d ( a3 ^2* b3 ^2*w0) )
c233
c233 . f a c t o r ( )
s 2 3 3 = -2*( p1^2+2*q1 ) * ( p2^2+2*q2 ) * ( p4^2+2*q4 ) * q1 * q2 * q3 ^2* q4
s233
#This i s t h e sum o f t h e terms with a3 ^2 and b3 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
2*a1^3*a2^3*a3^2*a4^3*b1*b2*b3^2*b4 + 2*a1*a2^3*a3^2*a4^3*b1^3*b2*b3^2*b4 +
2*a1^3*a2*a3^2*a4^3*b1*b2^3*b3^2*b4 + 2*a1*a2*a3^2*a4^3*b1^3*b2^3*b3^2*b4 +
2*a1^3*a2^3*a3^2*a4*b1*b2*b3^2*b4^3 + 2*a1*a2^3*a3^2*a4*b1^3*b2*b3^2*b4^3 +
2*a1^3*a2*a3^2*a4*b1*b2^3*b3^2*b4^3 + 2*a1*a2*a3^2*a4*b1^3*b2^3*b3^2*b4^3
2*(a1^2 + b1^2)*(a2^2 + b2^2)*(a4^2 + b4^2)*a1*a2*a3^2*a4*b1*b2*b3^2*b4
-2*(p1^2 + 2*q1)*(p2^2 + 2*q2)*(p4^2 + 2*q4)*q1*q2*q3^2*q4
c244=sum ( c2 . f i n d ( a4 ^2* b4 ^2*w0) )
c244
c244 . f a c t o r ( )
s 2 4 4 = -2*( p1^2+2*q1 ) * ( p2^2+2*q2 ) * ( p3^2+2*q3 ) * q1 * q2 * q3 * q4 ^2
s244
#This i s t h e sum o f t h e terms with a4 ^2 and b4 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
2*a1^3*a2^3*a3^3*a4^2*b1*b2*b3*b4^2 + 2*a1*a2^3*a3^3*a4^2*b1^3*b2*b3*b4^2 +
2*a1^3*a2*a3^3*a4^2*b1*b2^3*b3*b4^2 + 2*a1*a2*a3^3*a4^2*b1^3*b2^3*b3*b4^2 +
2*a1^3*a2^3*a3*a4^2*b1*b2*b3^3*b4^2 + 2*a1*a2^3*a3*a4^2*b1^3*b2*b3^3*b4^2 +
2*a1^3*a2*a3*a4^2*b1*b2^3*b3^3*b4^2 + 2*a1*a2*a3*a4^2*b1^3*b2^3*b3^3*b4^2
2*(a1^2 + b1^2)*(a2^2 + b2^2)*(a3^2 + b3^2)*a1*a2*a3*a4^2*b1*b2*b3*b4^2
-2*(p1^2 + 2*q1)*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q1*q2*q3*q4^2
b o o l ( c2==c211+c222+c233+c244 )
#This i s a check t o make s u r e a s we s e p e r a t e d t h e g r o u p s we used e v e r y term i n \
c2 once and o n l y once .
True
c3=sum ( f 1 1 . f i n d ( 3 *w0) )
c3
#This f i n d s a l l our terms i n f 1 1
t o g e t h e r . The gr oups we form
s e c o n d power .
3*a1^2*a2^2*a3^3*a4^3*b1^2*b2^2*b3*b4
3*a1^3*a2^2*a3^2*a4^3*b1*b2^2*b3^2*b4
3*a1^2*a2*a3^2*a4^3*b1^2*b2^3*b3^2*b4
3*a1^2*a2^3*a3^3*a4^2*b1^2*b2*b3*b4^2
3*a1*a2^2*a3^3*a4^2*b1^3*b2^2*b3*b4^2
3*a1^3*a2^3*a3^2*a4^2*b1*b2*b3^2*b4^2
3*a1^3*a2*a3^2*a4^2*b1*b2^3*b3^2*b4^2
3*a1^2*a2^3*a3*a4^2*b1^2*b2*b3^3*b4^2
3*a1*a2^2*a3*a4^2*b1^3*b2^2*b3^3*b4^2
3*a1^2*a2^2*a3^3*a4*b1^2*b2^2*b3*b4^3
3*a1^3*a2^2*a3^2*a4*b1*b2^2*b3^2*b4^3
3*a1^2*a2*a3^2*a4*b1^2*b2^3*b3^2*b4^3

t h a t have a 2 i n them and then adds them \
w i l l have two o f t h e f o u r a v a r i a b l e s t o t h e \
+
+
+
+
+
+
+
+
+
+
+
+

3*a1^2*a2^3*a3^2*a4^3*b1^2*b2*b3^2*b4
3*a1*a2^2*a3^2*a4^3*b1^3*b2^2*b3^2*b4
3*a1^2*a2^2*a3*a4^3*b1^2*b2^2*b3^3*b4
3*a1^3*a2^2*a3^3*a4^2*b1*b2^2*b3*b4^2
3*a1^2*a2*a3^3*a4^2*b1^2*b2^3*b3*b4^2
3*a1*a2^3*a3^2*a4^2*b1^3*b2*b3^2*b4^2
3*a1*a2*a3^2*a4^2*b1^3*b2^3*b3^2*b4^2
3*a1^3*a2^2*a3*a4^2*b1*b2^2*b3^3*b4^2
3*a1^2*a2*a3*a4^2*b1^2*b2^3*b3^3*b4^2
3*a1^2*a2^3*a3^2*a4*b1^2*b2*b3^2*b4^3
3*a1*a2^2*a3^2*a4*b1^3*b2^2*b3^2*b4^3
3*a1^2*a2^2*a3*a4*b1^2*b2^2*b3^3*b4^3

c312=sum ( c3 . f i n d ( a1 ^2* a2 ^2*w0) )
c312

9

+
+
+
+
+
+
+
+
+
+
+

c312 . f a c t o r ( )
s 3 1 2 =3*(p3^2+2*q3 ) * ( p4^2+2*q4 ) * q1 ^2* q2 ^2* q3 * q4
s312
#This i s t h e sum o f t h e terms with a1 ^2 and a2 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
3*a1^2*a2^2*a3^3*a4^3*b1^2*b2^2*b3*b4 + 3*a1^2*a2^2*a3*a4^3*b1^2*b2^2*b3^3*b4 +
3*a1^2*a2^2*a3^3*a4*b1^2*b2^2*b3*b4^3 + 3*a1^2*a2^2*a3*a4*b1^2*b2^2*b3^3*b4^3
3*(a3^2 + b3^2)*(a4^2 + b4^2)*a1^2*a2^2*a3*a4*b1^2*b2^2*b3*b4
3*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1^2*q2^2*q3*q4
c313=sum ( c3 . f i n d ( a1 ^2* a3 ^2*w0) )
c313
c313 . f a c t o r ( )
s 3 1 3 =3*(p2^2+2*q2 ) * ( p4^2+2*q4 ) * q1 ^2* q2 * q3 ^2* q4
s313
#This i s t h e sum o f t h e terms with a1 ^2 and a3 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
3*a1^2*a2^3*a3^2*a4^3*b1^2*b2*b3^2*b4 + 3*a1^2*a2*a3^2*a4^3*b1^2*b2^3*b3^2*b4 +
3*a1^2*a2^3*a3^2*a4*b1^2*b2*b3^2*b4^3 + 3*a1^2*a2*a3^2*a4*b1^2*b2^3*b3^2*b4^3
3*(a2^2 + b2^2)*(a4^2 + b4^2)*a1^2*a2*a3^2*a4*b1^2*b2*b3^2*b4
3*(p2^2 + 2*q2)*(p4^2 + 2*q4)*q1^2*q2*q3^2*q4
c314=sum ( c3 . f i n d ( a1 ^2* a4 ^2*w0) )
c314
c314 . f a c t o r ( )
s 3 1 4 =3*(p2^2+2*q2 ) * ( p3^2+2*q3 ) * q1 ^2* q2 * q3 * q4 ^2
s314
#This i s t h e sum o f t h e terms with a1 ^2 and a4 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
3*a1^2*a2^3*a3^3*a4^2*b1^2*b2*b3*b4^2 + 3*a1^2*a2*a3^3*a4^2*b1^2*b2^3*b3*b4^2 +
3*a1^2*a2^3*a3*a4^2*b1^2*b2*b3^3*b4^2 + 3*a1^2*a2*a3*a4^2*b1^2*b2^3*b3^3*b4^2
3*(a2^2 + b2^2)*(a3^2 + b3^2)*a1^2*a2*a3*a4^2*b1^2*b2*b3*b4^2
3*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q1^2*q2*q3*q4^2
c323=sum ( c3 . f i n d ( a2 ^2* a3 ^2*w0) )
c323
c323 . f a c t o r ( )
s 3 2 3 =3*(p1^2+2*q1 ) * ( p4^2+2*q4 ) * q1 * q2 ^2* q3 ^2* q4
s323
#This i s t h e sum o f t h e terms with a2 ^2 and a3 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
3*a1^3*a2^2*a3^2*a4^3*b1*b2^2*b3^2*b4 + 3*a1*a2^2*a3^2*a4^3*b1^3*b2^2*b3^2*b4 +
3*a1^3*a2^2*a3^2*a4*b1*b2^2*b3^2*b4^3 + 3*a1*a2^2*a3^2*a4*b1^3*b2^2*b3^2*b4^3
3*(a1^2 + b1^2)*(a4^2 + b4^2)*a1*a2^2*a3^2*a4*b1*b2^2*b3^2*b4
3*(p1^2 + 2*q1)*(p4^2 + 2*q4)*q1*q2^2*q3^2*q4
c324=sum ( c3 . f i n d ( a2 ^2* a4 ^2*w0) )
c324
c324 . f a c t o r ( )
s 3 2 4 =3*(p1^2+2*q1 ) * ( p3^2+2*q3 ) * q1 * q2 ^2* q3 * q4 ^2
s324
#This i s t h e sum o f t h e terms with a2 ^2 and a4 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
3*a1^3*a2^2*a3^3*a4^2*b1*b2^2*b3*b4^2 + 3*a1*a2^2*a3^3*a4^2*b1^3*b2^2*b3*b4^2 +
3*a1^3*a2^2*a3*a4^2*b1*b2^2*b3^3*b4^2 + 3*a1*a2^2*a3*a4^2*b1^3*b2^2*b3^3*b4^2
3*(a1^2 + b1^2)*(a3^2 + b3^2)*a1*a2^2*a3*a4^2*b1*b2^2*b3*b4^2
3*(p1^2 + 2*q1)*(p3^2 + 2*q3)*q1*q2^2*q3*q4^2
c334=sum ( c3 . f i n d ( a3 ^2* a4 ^2*w0) )
c334
c334 . f a c t o r ( )

10

s 3 3 4 =3*(p1^2+2*q1 ) * ( p2^2+2*q2 ) * q1 * q2 * q3 ^2* q4 ^2
s334
#This i s t h e sum o f t h e terms with a3 ^2 and a4 ^2 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
3*a1^3*a2^3*a3^2*a4^2*b1*b2*b3^2*b4^2 + 3*a1*a2^3*a3^2*a4^2*b1^3*b2*b3^2*b4^2 +
3*a1^3*a2*a3^2*a4^2*b1*b2^3*b3^2*b4^2 + 3*a1*a2*a3^2*a4^2*b1^3*b2^3*b3^2*b4^2
3*(a1^2 + b1^2)*(a2^2 + b2^2)*a1*a2*a3^2*a4^2*b1*b2*b3^2*b4^2
3*(p1^2 + 2*q1)*(p2^2 + 2*q2)*q1*q2*q3^2*q4^2
b o o l ( c3==c312+c313+c314+c323+c324+c334 )
#This i s a check t o make s u r e a s we s e p e r a t e d t h e g r o u p s we used e v e r y term i n \
c3 once and o n l y once .
True
c7=sum ( f 1 1 . f i n d ( 7 *w0) )
c7
#This f i n d s a l l our terms i n f 1 1 t h a t have a 7 i n them and then adds them \
t o g e t h e r . The gr oups we form w i l l have one o f t h e f o u r a v a r i a b l e s o r i t s \
c o r r e s p o n d i n g b v a r i a b l e t o t h e t h i r d power .
7*a1^2*a2^2*a3^2*a4^3*b1^2*b2^2*b3^2*b4 + 7*a1^2*a2^2*a3^3*a4^2*b1^2*b2^2*b3*b4^2 +
7*a1^2*a2^3*a3^2*a4^2*b1^2*b2*b3^2*b4^2 + 7*a1^3*a2^2*a3^2*a4^2*b1*b2^2*b3^2*b4^2 +
7*a1*a2^2*a3^2*a4^2*b1^3*b2^2*b3^2*b4^2 + 7*a1^2*a2*a3^2*a4^2*b1^2*b2^3*b3^2*b4^2 +
7*a1^2*a2^2*a3*a4^2*b1^2*b2^2*b3^3*b4^2 + 7*a1^2*a2^2*a3^2*a4*b1^2*b2^2*b3^2*b4^3
c71=sum ( c7 . f i n d ( a1 ^3*w0) )+sum ( c7 . f i n d ( b1 ^3*w0) )
c71
c71 . f a c t o r ( )
s 7 1 = -7*( p1^2 + 2* q1 ) * q1 * q2 ^2* q3 ^2* q4 ^2
s71
#This i s t h e sum o f t h e terms with a1 ^3 o r b1 ^3 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
7*a1^3*a2^2*a3^2*a4^2*b1*b2^2*b3^2*b4^2 + 7*a1*a2^2*a3^2*a4^2*b1^3*b2^2*b3^2*b4^2
7*(a1^2 + b1^2)*a1*a2^2*a3^2*a4^2*b1*b2^2*b3^2*b4^2
-7*(p1^2 + 2*q1)*q1*q2^2*q3^2*q4^2
c72=sum ( c7 . f i n d ( a2 ^3*w0) )+sum ( c7 . f i n d ( b2 ^3*w0) )
c72
c72 . f a c t o r ( )
s 7 2 = -7*( p2^2 + 2* q2 ) * q1 ^2* q2 * q3 ^2* q4 ^2
s72
#This i s t h e sum o f t h e terms with a2 ^3 o r b2 ^3 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
7*a1^2*a2^3*a3^2*a4^2*b1^2*b2*b3^2*b4^2 + 7*a1^2*a2*a3^2*a4^2*b1^2*b2^3*b3^2*b4^2
7*(a2^2 + b2^2)*a1^2*a2*a3^2*a4^2*b1^2*b2*b3^2*b4^2
-7*(p2^2 + 2*q2)*q1^2*q2*q3^2*q4^2
c73=sum ( c7 . f i n d ( a3 ^3*w0) )+sum ( c7 . f i n d ( b3 ^3*w0) )
c73
c73 . f a c t o r ( )
s 7 3 = -7*( p3^2 + 2* q3 ) * q1 ^2* q2 ^2* q3 * q4 ^2
s73
#This i s t h e sum o f t h e terms with a3 ^3 o r b3 ^3 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
7*a1^2*a2^2*a3^3*a4^2*b1^2*b2^2*b3*b4^2 + 7*a1^2*a2^2*a3*a4^2*b1^2*b2^2*b3^3*b4^2
7*(a3^2 + b3^2)*a1^2*a2^2*a3*a4^2*b1^2*b2^2*b3*b4^2
-7*(p3^2 + 2*q3)*q1^2*q2^2*q3*q4^2
c74=sum ( c7 . f i n d ( a4 ^3*w0) )+sum ( c7 . f i n d ( b4 ^3*w0) )
c74
c74 . f a c t o r ( )
s 7 4 = -7*( p4^2 + 2* q4 ) * q1 ^2* q2 ^2* q3 ^2* q4

11

s74
#This i s t h e sum o f t h e terms with a4 ^3 o r b4 ^3 , i t s f a c t o r i z a t i o n , and i t s \
substitution .
7*a1^2*a2^2*a3^2*a4^3*b1^2*b2^2*b3^2*b4 + 7*a1^2*a2^2*a3^2*a4*b1^2*b2^2*b3^2*b4^3
7*(a4^2 + b4^2)*a1^2*a2^2*a3^2*a4*b1^2*b2^2*b3^2*b4
-7*(p4^2 + 2*q4)*q1^2*q2^2*q3^2*q4
b o o l ( c7==c71+c72+c73+c74 )
#This i s a check t o make s u r e a s we s e p e r a t e d t h e g r o u p s we used e v e r y term i n \
c7 once and o n l y once .
True
c9=sum ( f 1 1 . f i n d ( 9 *w0) )
c9
s 9 =9*q1 ^2* q2 ^2* q3 ^2* q4 ^2
s9
#This i s t h e term form f 1 1 with a 9 i n i t and i t s s u b s t i t u t i o n .
9*a1^2*a2^2*a3^2*a4^2*b1^2*b2^2*b3^2*b4^2
9*q1^2*q2^2*q3^2*q4^2
c1=sum ( f 1 1 . f i n d ( 1 *w0) ) - c2 - c3 - c7 - c9
c1
#This f i n d s a l l our terms i n f 1 1 t h a t have a 1 i n them and then adds them \
t o g e t h e r . The f i r s t grou ps we form w i l l have t h r e e o f t h e f o u r a v a r i a b l e s \
t o t h e se c o n d power .
a1^2*a2^2*a3^3*a4^4*b1^2*b2^2*b3 + a1^2*a2^3*a3^2*a4^4*b1^2*b2*b3^2 +
a1^3*a2^2*a3^2*a4^4*b1*b2^2*b3^2 + a1^2*a2^2*a3^2*a4^4*b1^2*b2^2*b3^2 +
a1*a2^2*a3^2*a4^4*b1^3*b2^2*b3^2 + a1^2*a2*a3^2*a4^4*b1^2*b2^3*b3^2 +
a1^2*a2^2*a3*a4^4*b1^2*b2^2*b3^3 + a1^2*a2^2*a3^4*a4^3*b1^2*b2^2*b4 +
a1^3*a2^3*a3^3*a4^3*b1*b2*b3*b4 + a1*a2^3*a3^3*a4^3*b1^3*b2*b3*b4 +
a1^3*a2*a3^3*a4^3*b1*b2^3*b3*b4 + a1*a2*a3^3*a4^3*b1^3*b2^3*b3*b4 +
a1^2*a2^4*a3^2*a4^3*b1^2*b3^2*b4 + a1^4*a2^2*a3^2*a4^3*b2^2*b3^2*b4 +
a2^2*a3^2*a4^3*b1^4*b2^2*b3^2*b4 + a1^2*a3^2*a4^3*b1^2*b2^4*b3^2*b4 +
a1^3*a2^3*a3*a4^3*b1*b2*b3^3*b4 + a1*a2^3*a3*a4^3*b1^3*b2*b3^3*b4 +
a1^3*a2*a3*a4^3*b1*b2^3*b3^3*b4 + a1*a2*a3*a4^3*b1^3*b2^3*b3^3*b4 +
a1^2*a2^2*a4^3*b1^2*b2^2*b3^4*b4 + a1^2*a2^3*a3^4*a4^2*b1^2*b2*b4^2 +
a1^3*a2^2*a3^4*a4^2*b1*b2^2*b4^2 + a1^2*a2^2*a3^4*a4^2*b1^2*b2^2*b4^2 +
a1*a2^2*a3^4*a4^2*b1^3*b2^2*b4^2 + a1^2*a2*a3^4*a4^2*b1^2*b2^3*b4^2 +
a1^2*a2^4*a3^3*a4^2*b1^2*b3*b4^2 + a1^4*a2^2*a3^3*a4^2*b2^2*b3*b4^2 +
a2^2*a3^3*a4^2*b1^4*b2^2*b3*b4^2 + a1^2*a3^3*a4^2*b1^2*b2^4*b3*b4^2 +
a1^3*a2^4*a3^2*a4^2*b1*b3^2*b4^2 + a1^2*a2^4*a3^2*a4^2*b1^2*b3^2*b4^2 +
a1*a2^4*a3^2*a4^2*b1^3*b3^2*b4^2 + a1^4*a2^3*a3^2*a4^2*b2*b3^2*b4^2 +
a2^3*a3^2*a4^2*b1^4*b2*b3^2*b4^2 + a1^4*a2^2*a3^2*a4^2*b2^2*b3^2*b4^2 +
a2^2*a3^2*a4^2*b1^4*b2^2*b3^2*b4^2 + a1^4*a2*a3^2*a4^2*b2^3*b3^2*b4^2 +
a2*a3^2*a4^2*b1^4*b2^3*b3^2*b4^2 + a1^3*a3^2*a4^2*b1*b2^4*b3^2*b4^2 +
a1^2*a3^2*a4^2*b1^2*b2^4*b3^2*b4^2 + a1*a3^2*a4^2*b1^3*b2^4*b3^2*b4^2 +
a1^2*a2^4*a3*a4^2*b1^2*b3^3*b4^2 + a1^4*a2^2*a3*a4^2*b2^2*b3^3*b4^2 +
a2^2*a3*a4^2*b1^4*b2^2*b3^3*b4^2 + a1^2*a3*a4^2*b1^2*b2^4*b3^3*b4^2 +
a1^2*a2^3*a4^2*b1^2*b2*b3^4*b4^2 + a1^3*a2^2*a4^2*b1*b2^2*b3^4*b4^2 +
a1^2*a2^2*a4^2*b1^2*b2^2*b3^4*b4^2 + a1*a2^2*a4^2*b1^3*b2^2*b3^4*b4^2 +
a1^2*a2*a4^2*b1^2*b2^3*b3^4*b4^2 + a1^2*a2^2*a3^4*a4*b1^2*b2^2*b4^3 +
a1^3*a2^3*a3^3*a4*b1*b2*b3*b4^3 + a1*a2^3*a3^3*a4*b1^3*b2*b3*b4^3 +
a1^3*a2*a3^3*a4*b1*b2^3*b3*b4^3 + a1*a2*a3^3*a4*b1^3*b2^3*b3*b4^3 +
a1^2*a2^4*a3^2*a4*b1^2*b3^2*b4^3 + a1^4*a2^2*a3^2*a4*b2^2*b3^2*b4^3 +
a2^2*a3^2*a4*b1^4*b2^2*b3^2*b4^3 + a1^2*a3^2*a4*b1^2*b2^4*b3^2*b4^3 +
a1^3*a2^3*a3*a4*b1*b2*b3^3*b4^3 + a1*a2^3*a3*a4*b1^3*b2*b3^3*b4^3 +
a1^3*a2*a3*a4*b1*b2^3*b3^3*b4^3 + a1*a2*a3*a4*b1^3*b2^3*b3^3*b4^3 +
a1^2*a2^2*a4*b1^2*b2^2*b3^4*b4^3 + a1^2*a2^2*a3^3*b1^2*b2^2*b3*b4^4 +
a1^2*a2^3*a3^2*b1^2*b2*b3^2*b4^4 + a1^3*a2^2*a3^2*b1*b2^2*b3^2*b4^4 +
a1^2*a2^2*a3^2*b1^2*b2^2*b3^2*b4^4 + a1*a2^2*a3^2*b1^3*b2^2*b3^2*b4^4 +
a1^2*a2*a3^2*b1^2*b2^3*b3^2*b4^4 + a1^2*a2^2*a3*b1^2*b2^2*b3^3*b4^4

12

c11=sum ( c1 . f i n d ( a2 ^2* a3 ^2* a4 ^2*w0) )
c11
c11 . f a c t o r ( )
s 1 1 =(( p1^2 + 2* q1 ) ^2 - 2* q1 ^2) * q2 ^2* q3 ^2* q4 ^2
s11
#This i s t h e sum o f t h e terms with a2 ^2 , a3 ^2 , and a4 ^2 , i t s f a c t o r i z a t i o n , and\
its substitution .
a1^4*a2^2*a3^2*a4^2*b2^2*b3^2*b4^2 + a2^2*a3^2*a4^2*b1^4*b2^2*b3^2*b4^2
(a1^4 + b1^4)*a2^2*a3^2*a4^2*b2^2*b3^2*b4^2
((p1^2 + 2*q1)^2 - 2*q1^2)*q2^2*q3^2*q4^2
c12=sum ( c1 . f i n d ( a1 ^2* a3 ^2* a4 ^2*w0) )
c12
c12 . f a c t o r ( )
s 1 2 =(( p2^2 + 2* q2 ) ^2 - 2* q2 ^2) * q1 ^2* q3 ^2* q4 ^2
s12
#This i s t h e sum o f t h e terms with a1 ^2 , a3 ^2 , and a4 ^2 , i t s f a c t o r i z a t i o n , and\
its substitution .
a1^2*a2^4*a3^2*a4^2*b1^2*b3^2*b4^2 + a1^2*a3^2*a4^2*b1^2*b2^4*b3^2*b4^2
(a2^4 + b2^4)*a1^2*a3^2*a4^2*b1^2*b3^2*b4^2
((p2^2 + 2*q2)^2 - 2*q2^2)*q1^2*q3^2*q4^2
c13=sum ( c1 . f i n d ( a1 ^2* a2 ^2* a4 ^2*w0) )
c13
c13 . f a c t o r ( )
s 1 3 =(( p3^2 + 2* q3 ) ^2 - 2* q3 ^2) * q1 ^2* q2 ^2* q4 ^2
s13
#This i s t h e sum o f t h e terms with a1 ^2 , a2 ^2 , and a4 ^2 , i t s f a c t o r i z a t i o n , and\
its substitution .
a1^2*a2^2*a3^4*a4^2*b1^2*b2^2*b4^2 + a1^2*a2^2*a4^2*b1^2*b2^2*b3^4*b4^2
(a3^4 + b3^4)*a1^2*a2^2*a4^2*b1^2*b2^2*b4^2
((p3^2 + 2*q3)^2 - 2*q3^2)*q1^2*q2^2*q4^2
c14=sum ( c1 . f i n d ( a1 ^2* a2 ^2* a3 ^2*w0) )
c14
c14 . f a c t o r ( )
s 1 4 =(( p4^2 + 2* q4 ) ^2 - 2* q4 ^2) * q1 ^2* q2 ^2* q3 ^2
s14
#This i s t h e sum o f t h e terms with a1 ^2 , a2 ^2 , and a3 ^2 , i t s f a c t o r i z a t i o n , and\
its substitution .
a1^2*a2^2*a3^2*a4^4*b1^2*b2^2*b3^2 + a1^2*a2^2*a3^2*b1^2*b2^2*b3^2*b4^4
(a4^4 + b4^4)*a1^2*a2^2*a3^2*b1^2*b2^2*b3^2
((p4^2 + 2*q4)^2 - 2*q4^2)*q1^2*q2^2*q3^2
c01=c1 - c11 - c12 - c13 - c14
c01
#This g e t s a l l t h e terms from c1 t h a t have not been a s s i g n e d t o a group . The \
g ro u p s we form w i l l have two o f t h e f o u r a v a r i a b l e s t o t h e s e c o n d power and\
e i t h e r a t h i r d a v a r i a b l e or i t s corresponding b v a r i a b l e to the t h i r d \
power .
a1^2*a2^2*a3^3*a4^4*b1^2*b2^2*b3 + a1^2*a2^3*a3^2*a4^4*b1^2*b2*b3^2 +
a1^3*a2^2*a3^2*a4^4*b1*b2^2*b3^2 + a1*a2^2*a3^2*a4^4*b1^3*b2^2*b3^2 +
a1^2*a2*a3^2*a4^4*b1^2*b2^3*b3^2 + a1^2*a2^2*a3*a4^4*b1^2*b2^2*b3^3 +
a1^2*a2^2*a3^4*a4^3*b1^2*b2^2*b4 + a1^3*a2^3*a3^3*a4^3*b1*b2*b3*b4 +
a1*a2^3*a3^3*a4^3*b1^3*b2*b3*b4 + a1^3*a2*a3^3*a4^3*b1*b2^3*b3*b4 +
a1*a2*a3^3*a4^3*b1^3*b2^3*b3*b4 + a1^2*a2^4*a3^2*a4^3*b1^2*b3^2*b4 +
a1^4*a2^2*a3^2*a4^3*b2^2*b3^2*b4 + a2^2*a3^2*a4^3*b1^4*b2^2*b3^2*b4 +
a1^2*a3^2*a4^3*b1^2*b2^4*b3^2*b4 + a1^3*a2^3*a3*a4^3*b1*b2*b3^3*b4 +
a1*a2^3*a3*a4^3*b1^3*b2*b3^3*b4 + a1^3*a2*a3*a4^3*b1*b2^3*b3^3*b4 +
a1*a2*a3*a4^3*b1^3*b2^3*b3^3*b4 + a1^2*a2^2*a4^3*b1^2*b2^2*b3^4*b4 +
a1^2*a2^3*a3^4*a4^2*b1^2*b2*b4^2 + a1^3*a2^2*a3^4*a4^2*b1*b2^2*b4^2 +

13

a1*a2^2*a3^4*a4^2*b1^3*b2^2*b4^2 + a1^2*a2*a3^4*a4^2*b1^2*b2^3*b4^2 +
a1^2*a2^4*a3^3*a4^2*b1^2*b3*b4^2 + a1^4*a2^2*a3^3*a4^2*b2^2*b3*b4^2 +
a2^2*a3^3*a4^2*b1^4*b2^2*b3*b4^2 + a1^2*a3^3*a4^2*b1^2*b2^4*b3*b4^2 +
a1^3*a2^4*a3^2*a4^2*b1*b3^2*b4^2 + a1*a2^4*a3^2*a4^2*b1^3*b3^2*b4^2 +
a1^4*a2^3*a3^2*a4^2*b2*b3^2*b4^2 + a2^3*a3^2*a4^2*b1^4*b2*b3^2*b4^2 +
a1^4*a2*a3^2*a4^2*b2^3*b3^2*b4^2 + a2*a3^2*a4^2*b1^4*b2^3*b3^2*b4^2 +
a1^3*a3^2*a4^2*b1*b2^4*b3^2*b4^2 + a1*a3^2*a4^2*b1^3*b2^4*b3^2*b4^2 +
a1^2*a2^4*a3*a4^2*b1^2*b3^3*b4^2 + a1^4*a2^2*a3*a4^2*b2^2*b3^3*b4^2 +
a2^2*a3*a4^2*b1^4*b2^2*b3^3*b4^2 + a1^2*a3*a4^2*b1^2*b2^4*b3^3*b4^2 +
a1^2*a2^3*a4^2*b1^2*b2*b3^4*b4^2 + a1^3*a2^2*a4^2*b1*b2^2*b3^4*b4^2 +
a1*a2^2*a4^2*b1^3*b2^2*b3^4*b4^2 + a1^2*a2*a4^2*b1^2*b2^3*b3^4*b4^2 +
a1^2*a2^2*a3^4*a4*b1^2*b2^2*b4^3 + a1^3*a2^3*a3^3*a4*b1*b2*b3*b4^3 +
a1*a2^3*a3^3*a4*b1^3*b2*b3*b4^3 + a1^3*a2*a3^3*a4*b1*b2^3*b3*b4^3 +
a1*a2*a3^3*a4*b1^3*b2^3*b3*b4^3 + a1^2*a2^4*a3^2*a4*b1^2*b3^2*b4^3 +
a1^4*a2^2*a3^2*a4*b2^2*b3^2*b4^3 + a2^2*a3^2*a4*b1^4*b2^2*b3^2*b4^3 +
a1^2*a3^2*a4*b1^2*b2^4*b3^2*b4^3 + a1^3*a2^3*a3*a4*b1*b2*b3^3*b4^3 +
a1*a2^3*a3*a4*b1^3*b2*b3^3*b4^3 + a1^3*a2*a3*a4*b1*b2^3*b3^3*b4^3 +
a1*a2*a3*a4*b1^3*b2^3*b3^3*b4^3 + a1^2*a2^2*a4*b1^2*b2^2*b3^4*b4^3 +
a1^2*a2^2*a3^3*b1^2*b2^2*b3*b4^4 + a1^2*a2^3*a3^2*b1^2*b2*b3^2*b4^4 +
a1^3*a2^2*a3^2*b1*b2^2*b3^2*b4^4 + a1*a2^2*a3^2*b1^3*b2^2*b3^2*b4^4 +
a1^2*a2*a3^2*b1^2*b2^3*b3^2*b4^4 + a1^2*a2^2*a3*b1^2*b2^2*b3^3*b4^4
c01123=sum ( c01 . f i n d ( a1 ^2* a2 ^2* a3 ^3*w0) )+sum ( c01 . f i n d ( a1 ^2* a2 ^2* b3 ^3*w0) )
c01123
c01123 . f a c t o r ( )
s0 1 1 2 3 = -(( p4^2 + 2* q4 ) ^2 - 2* q4 ^2) * ( p3^2 + 2* q3 ) * q1 ^2* q2 ^2* q3
s0 1 1 2 3
#This i s t h e sum o f t h e terms with a1 ^2 , a2 ^2 , and e i t h e r a3 ^3 o r b3 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^2*a2^2*a3^3*a4^4*b1^2*b2^2*b3 + a1^2*a2^2*a3*a4^4*b1^2*b2^2*b3^3 +
a1^2*a2^2*a3^3*b1^2*b2^2*b3*b4^4 + a1^2*a2^2*a3*b1^2*b2^2*b3^3*b4^4
(a4^4 + b4^4)*(a3^2 + b3^2)*a1^2*a2^2*a3*b1^2*b2^2*b3
-((p4^2 + 2*q4)^2 - 2*q4^2)*(p3^2 + 2*q3)*q1^2*q2^2*q3
c01124=sum ( c01 . f i n d ( a1 ^2* a2 ^2* a4 ^3*w0) )+sum ( c01 . f i n d ( a1 ^2* a2 ^2* b4 ^3*w0) )
c01124
c01124 . f a c t o r ( )
s0 1 1 2 4 = -(( p3^2 + 2* q3 ) ^2 - 2* q3 ^2) * ( p4^2 + 2* q4 ) * q1 ^2* q2 ^2* q4
s0 1 1 2 4
#This i s t h e sum o f t h e terms with a1 ^2 , a2 ^2 , and e i t h e r a4 ^3 o r b4 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^2*a2^2*a3^4*a4^3*b1^2*b2^2*b4 + a1^2*a2^2*a4^3*b1^2*b2^2*b3^4*b4 +
a1^2*a2^2*a3^4*a4*b1^2*b2^2*b4^3 + a1^2*a2^2*a4*b1^2*b2^2*b3^4*b4^3
(a3^4 + b3^4)*(a4^2 + b4^2)*a1^2*a2^2*a4*b1^2*b2^2*b4
-((p3^2 + 2*q3)^2 - 2*q3^2)*(p4^2 + 2*q4)*q1^2*q2^2*q4
c01132=sum ( c01 . f i n d ( a1 ^2* a3 ^2* a2 ^3*w0) )+sum ( c01 . f i n d ( a1 ^2* a3 ^2* b2 ^3*w0) )
c01132
c01132 . f a c t o r ( )
s0 1 1 3 2 = -(( p4^2 + 2* q4 ) ^2 - 2* q4 ^2) * ( p2^2 + 2* q2 ) * q1 ^2* q2 * q3 ^2
s0 1 1 3 2
#This i s t h e sum o f t h e terms with a1 ^2 , a3 ^2 , and e i t h e r a2 ^3 o r b2 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^2*a2^3*a3^2*a4^4*b1^2*b2*b3^2 + a1^2*a2*a3^2*a4^4*b1^2*b2^3*b3^2 +
a1^2*a2^3*a3^2*b1^2*b2*b3^2*b4^4 + a1^2*a2*a3^2*b1^2*b2^3*b3^2*b4^4
(a4^4 + b4^4)*(a2^2 + b2^2)*a1^2*a2*a3^2*b1^2*b2*b3^2
-((p4^2 + 2*q4)^2 - 2*q4^2)*(p2^2 + 2*q2)*q1^2*q2*q3^2
c01134=sum ( c01 . f i n d ( a1 ^2* a3 ^2* a4 ^3*w0) )+sum ( c01 . f i n d ( a1 ^2* a3 ^2* b4 ^3*w0) )
c01134
c01134 . f a c t o r ( )

14

s0 1 1 3 4 = -(( p2^2 + 2* q2 ) ^2 - 2* q2 ^2) * ( p4^2 + 2* q4 ) * q1 ^2* q3 ^2* q4
s0 1 1 3 4
#This i s t h e sum o f t h e terms with a1 ^2 , a3 ^2 , and e i t h e r a4 ^3 o r b4 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^2*a2^4*a3^2*a4^3*b1^2*b3^2*b4 + a1^2*a3^2*a4^3*b1^2*b2^4*b3^2*b4 +
a1^2*a2^4*a3^2*a4*b1^2*b3^2*b4^3 + a1^2*a3^2*a4*b1^2*b2^4*b3^2*b4^3
(a2^4 + b2^4)*(a4^2 + b4^2)*a1^2*a3^2*a4*b1^2*b3^2*b4
-((p2^2 + 2*q2)^2 - 2*q2^2)*(p4^2 + 2*q4)*q1^2*q3^2*q4
c01142=sum ( c01 . f i n d ( a1 ^2* a4 ^2* a2 ^3*w0) )+sum ( c01 . f i n d ( a1 ^2* a4 ^2* b2 ^3*w0) )
c01142
c01142 . f a c t o r ( )
s0 1 1 4 2 = -(( p3^2 + 2* q3 ) ^2 - 2* q3 ^2) * ( p2^2 + 2* q2 ) * q1 ^2* q2 * q4 ^2
s0 1 1 4 2
#This i s t h e sum o f t h e terms with a1 ^2 , a4 ^2 , and e i t h e r a2 ^3 o r b2 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^2*a2^3*a3^4*a4^2*b1^2*b2*b4^2 + a1^2*a2*a3^4*a4^2*b1^2*b2^3*b4^2 +
a1^2*a2^3*a4^2*b1^2*b2*b3^4*b4^2 + a1^2*a2*a4^2*b1^2*b2^3*b3^4*b4^2
(a3^4 + b3^4)*(a2^2 + b2^2)*a1^2*a2*a4^2*b1^2*b2*b4^2
-((p3^2 + 2*q3)^2 - 2*q3^2)*(p2^2 + 2*q2)*q1^2*q2*q4^2
c01143=sum ( c01 . f i n d ( a1 ^2* a4 ^2* a3 ^3*w0) )+sum ( c01 . f i n d ( a1 ^2* a4 ^2* b3 ^3*w0) )
c01143
c01143 . f a c t o r ( )
s0 1 1 4 3 = -(( p2^2 + 2* q2 ) ^2 - 2* q2 ^2) * ( p3^2 + 2* q3 ) * q1 ^2* q3 * q4 ^2
s0 1 1 4 3
#This i s t h e sum o f t h e terms with a1 ^2 , a4 ^2 , and e i t h e r a3 ^3 o r b3 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^2*a2^4*a3^3*a4^2*b1^2*b3*b4^2 + a1^2*a3^3*a4^2*b1^2*b2^4*b3*b4^2 +
a1^2*a2^4*a3*a4^2*b1^2*b3^3*b4^2 + a1^2*a3*a4^2*b1^2*b2^4*b3^3*b4^2
(a2^4 + b2^4)*(a3^2 + b3^2)*a1^2*a3*a4^2*b1^2*b3*b4^2
-((p2^2 + 2*q2)^2 - 2*q2^2)*(p3^2 + 2*q3)*q1^2*q3*q4^2
c01231=sum ( c01 . f i n d ( a2 ^2* a3 ^2* a1 ^3*w0) )+sum ( c01 . f i n d ( a2 ^2* a3 ^2* b1 ^3*w0) )
c01231
c01231 . f a c t o r ( )
s0 1 2 3 1 = -(( p4^2 + 2* q4 ) ^2 - 2* q4 ^2) * ( p1^2 + 2* q1 ) * q1 * q2 ^2* q3 ^2
s0 1 2 3 1
#This i s t h e sum o f t h e terms with a2 ^2 , a3 ^2 , and e i t h e r a1 ^3 o r b1 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^3*a2^2*a3^2*a4^4*b1*b2^2*b3^2 + a1*a2^2*a3^2*a4^4*b1^3*b2^2*b3^2 +
a1^3*a2^2*a3^2*b1*b2^2*b3^2*b4^4 + a1*a2^2*a3^2*b1^3*b2^2*b3^2*b4^4
(a4^4 + b4^4)*(a1^2 + b1^2)*a1*a2^2*a3^2*b1*b2^2*b3^2
-((p4^2 + 2*q4)^2 - 2*q4^2)*(p1^2 + 2*q1)*q1*q2^2*q3^2
c01234=sum ( c01 . f i n d ( a2 ^2* a3 ^2* a4 ^3*w0) )+sum ( c01 . f i n d ( a2 ^2* a3 ^2* b4 ^3*w0) )
c01234
c01234 . f a c t o r ( )
s0 1 2 3 4 = -(( p1^2 + 2* q1 ) ^2 - 2* q1 ^2) * ( p4^2 + 2* q4 ) * q2 ^2* q3 ^2* q4
s0 1 2 3 4
#This i s t h e sum o f t h e terms with a2 ^2 , a3 ^2 , and e i t h e r a4 ^3 o r b4 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^4*a2^2*a3^2*a4^3*b2^2*b3^2*b4 + a2^2*a3^2*a4^3*b1^4*b2^2*b3^2*b4 +
a1^4*a2^2*a3^2*a4*b2^2*b3^2*b4^3 + a2^2*a3^2*a4*b1^4*b2^2*b3^2*b4^3
(a1^4 + b1^4)*(a4^2 + b4^2)*a2^2*a3^2*a4*b2^2*b3^2*b4
-((p1^2 + 2*q1)^2 - 2*q1^2)*(p4^2 + 2*q4)*q2^2*q3^2*q4
c01241=sum ( c01 . f i n d ( a2 ^2* a4 ^2* a1 ^3*w0) )+sum ( c01 . f i n d ( a2 ^2* a4 ^2* b1 ^3*w0) )
c01241
c01241 . f a c t o r ( )
s0 1 2 4 1 = -(( p3^2 + 2* q3 ) ^2 - 2* q3 ^2) * ( p1^2 + 2* q1 ) * q1 * q2 ^2* q4 ^2

15

s0 1 2 4 1
#This i s t h e sum o f t h e terms with a2 ^2 , a4 ^2 , and e i t h e r a1 ^3 o r b1 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^3*a2^2*a3^4*a4^2*b1*b2^2*b4^2 + a1*a2^2*a3^4*a4^2*b1^3*b2^2*b4^2 +
a1^3*a2^2*a4^2*b1*b2^2*b3^4*b4^2 + a1*a2^2*a4^2*b1^3*b2^2*b3^4*b4^2
(a3^4 + b3^4)*(a1^2 + b1^2)*a1*a2^2*a4^2*b1*b2^2*b4^2
-((p3^2 + 2*q3)^2 - 2*q3^2)*(p1^2 + 2*q1)*q1*q2^2*q4^2
c01243=sum ( c01 . f i n d ( a2 ^2* a4 ^2* a3 ^3*w0) )+sum ( c01 . f i n d ( a2 ^2* a4 ^2* b3 ^3*w0) )
c01243
c01243 . f a c t o r ( )
s0 1 2 4 3 = -(( p1^2 + 2* q1 ) ^2 - 2* q1 ^2) * ( p3^2 + 2* q3 ) * q2 ^2* q3 * q4 ^2
s0 1 2 4 3
#This i s t h e sum o f t h e terms with a2 ^2 , a4 ^2 , and e i t h e r a3 ^3 o r b3 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^4*a2^2*a3^3*a4^2*b2^2*b3*b4^2 + a2^2*a3^3*a4^2*b1^4*b2^2*b3*b4^2 +
a1^4*a2^2*a3*a4^2*b2^2*b3^3*b4^2 + a2^2*a3*a4^2*b1^4*b2^2*b3^3*b4^2
(a1^4 + b1^4)*(a3^2 + b3^2)*a2^2*a3*a4^2*b2^2*b3*b4^2
-((p1^2 + 2*q1)^2 - 2*q1^2)*(p3^2 + 2*q3)*q2^2*q3*q4^2
c01341=sum ( c01 . f i n d ( a3 ^2* a4 ^2* a1 ^3*w0) )+sum ( c01 . f i n d ( a3 ^2* a4 ^2* b1 ^3*w0) )
c01341
c01341 . f a c t o r ( )
s0 1 3 4 1 = -(( p2^2 + 2* q2 ) ^2 - 2* q2 ^2) * ( p1^2 + 2* q1 ) * q1 * q3 ^2* q4 ^2
s0 1 3 4 1
#This i s t h e sum o f t h e terms with a3 ^2 , a4 ^2 , and e i t h e r a1 ^3 o r b1 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^3*a2^4*a3^2*a4^2*b1*b3^2*b4^2 + a1*a2^4*a3^2*a4^2*b1^3*b3^2*b4^2 +
a1^3*a3^2*a4^2*b1*b2^4*b3^2*b4^2 + a1*a3^2*a4^2*b1^3*b2^4*b3^2*b4^2
(a2^4 + b2^4)*(a1^2 + b1^2)*a1*a3^2*a4^2*b1*b3^2*b4^2
-((p2^2 + 2*q2)^2 - 2*q2^2)*(p1^2 + 2*q1)*q1*q3^2*q4^2
c01342=sum ( c01 . f i n d ( a3 ^2* a4 ^2* a2 ^3*w0) )+sum ( c01 . f i n d ( a3 ^2* a4 ^2* b2 ^3*w0) )
c01342
c01342 . f a c t o r ( )
s0 1 3 4 2 = -(( p1^2 + 2* q1 ) ^2 - 2* q1 ^2) * ( p2^2 + 2* q2 ) * q2 * q3 ^2* q4 ^2
s0 1 3 4 2
#This i s t h e sum o f t h e terms with a3 ^2 , a4 ^2 , and e i t h e r a2 ^3 o r b2 ^3 , i t s \
f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^4*a2^3*a3^2*a4^2*b2*b3^2*b4^2 + a2^3*a3^2*a4^2*b1^4*b2*b3^2*b4^2 +
a1^4*a2*a3^2*a4^2*b2^3*b3^2*b4^2 + a2*a3^2*a4^2*b1^4*b2^3*b3^2*b4^2
(a1^4 + b1^4)*(a2^2 + b2^2)*a2*a3^2*a4^2*b2*b3^2*b4^2
-((p1^2 + 2*q1)^2 - 2*q1^2)*(p2^2 + 2*q2)*q2*q3^2*q4^2
c001=c01 - c01123 - c01124 - c01132 - c01134 - c01142 - c01143 - c01231 - c01234 - c01241 - c01243 - \
c01341 - c01342
c001
c001 . f a c t o r ( )
s 0 0 1 =(p1^2 + 2* q1 ) * ( p2^2 + 2* q2 ) * ( p3^2 + 2* q3 ) * ( p4^2 + 2* q4 ) * q1 * q2 * q3 * q4
s001
#This i s t h e sum o f a l l t h e terms from c01 t h a t have not been a s s i g n e d t o a \
group , i t s f a c t o r i z a t i o n , and i t s s u b s t i t u t i o n .
a1^3*a2^3*a3^3*a4^3*b1*b2*b3*b4 + a1*a2^3*a3^3*a4^3*b1^3*b2*b3*b4 +
a1^3*a2*a3^3*a4^3*b1*b2^3*b3*b4 + a1*a2*a3^3*a4^3*b1^3*b2^3*b3*b4 +
a1^3*a2^3*a3*a4^3*b1*b2*b3^3*b4 + a1*a2^3*a3*a4^3*b1^3*b2*b3^3*b4 +
a1^3*a2*a3*a4^3*b1*b2^3*b3^3*b4 + a1*a2*a3*a4^3*b1^3*b2^3*b3^3*b4 +
a1^3*a2^3*a3^3*a4*b1*b2*b3*b4^3 + a1*a2^3*a3^3*a4*b1^3*b2*b3*b4^3 +
a1^3*a2*a3^3*a4*b1*b2^3*b3*b4^3 + a1*a2*a3^3*a4*b1^3*b2^3*b3*b4^3 +
a1^3*a2^3*a3*a4*b1*b2*b3^3*b4^3 + a1*a2^3*a3*a4*b1^3*b2*b3^3*b4^3 +
a1^3*a2*a3*a4*b1*b2^3*b3^3*b4^3 + a1*a2*a3*a4*b1^3*b2^3*b3^3*b4^3
(a1^2 + b1^2)*(a2^2 + b2^2)*(a3^2 + b3^2)*(a4^2 + b4^2)*a1*a2*a3*a4*b1*b2*b3*b4

16

(p1^2 + 2*q1)*(p2^2 + 2*q2)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1*q2*q3*q4
b o o l ( c11+c12+c13+c14+c01123+c01124+c01132+c01134+c01142+c01143+c01231+c01234+\
c01241+c01243+c01341+c01342+c001==c1 )
#This i s a check t o make s u r e a s we s e p e r a t e d t h e g r o u p s we used e v e r y term i n \
c1 once and o n l y once .
True
b o o l ( c11+c12+c13+c14+c01123+c01124+c01132+c01134+c01142+c01143+c01231+c01234+\
c01241+c01243+c01341+c01342+c001+c211+c222+c233+c244+c312+c313+c314+c323+\
c324+c334+c71+c72+c73+c74+c9==f 1 1 )
#This i s a check t o make s u r e a s we s e p e r a t e d t h e g r o u p s we used e v e r y term i n \
f 1 1 once and o n l y once .
True
l=s * ( s 1 1+s 1 2+s 1 3+s 1 4+s 0 1 1 2 3+s 0 1 1 2 4+s 0 1 1 3 2+s 0 1 1 3 4+s 0 1 1 4 2+s 0 1 1 4 3+s 0 1 2 3 1+s 0 1 2 3 4+\
s 0 1 2 4 1+s 0 12 4 3+s 0 13 4 1+s 0 1 34 2+s 0 0 1+s 2 1 1+s 2 2 2+s 2 3 3+s 2 4 4+s 3 1 2+s 3 1 3+s 3 1 4+s 3 2 3+\
s 3 2 4+s 3 3 4+s 7 1+s 7 2+s 7 3+s 7 4+s 9 )
l
#This l i n e b r i n g s a l l t h e group s u b s t i t u t i o n s back i n t o one e x p r e s s i o n .
((p1^2 + 2*q1)*(p2^2 + 2*q2)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1*q2*q3*q4 - 2*(p2^2 +
2*q2)*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1^2*q2*q3*q4 - 2*(p1^2 + 2*q1)*(p3^2 + 2*q3)*(p4^2 +
2*q4)*q1*q2^2*q3*q4 + 3*(p3^2 + 2*q3)*(p4^2 + 2*q4)*q1^2*q2^2*q3*q4 - 2*(p1^2 +
2*q1)*(p2^2 + 2*q2)*(p4^2 + 2*q4)*q1*q2*q3^2*q4 + 3*(p2^2 + 2*q2)*(p4^2 +
2*q4)*q1^2*q2*q3^2*q4 + 3*(p1^2 + 2*q1)*(p4^2 + 2*q4)*q1*q2^2*q3^2*q4 - 7*(p4^2 +
2*q4)*q1^2*q2^2*q3^2*q4 - 2*(p1^2 + 2*q1)*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q1*q2*q3*q4^2 +
3*(p2^2 + 2*q2)*(p3^2 + 2*q3)*q1^2*q2*q3*q4^2 + 3*(p1^2 + 2*q1)*(p3^2 +
2*q3)*q1*q2^2*q3*q4^2 - 7*(p3^2 + 2*q3)*q1^2*q2^2*q3*q4^2 + 3*(p1^2 + 2*q1)*(p2^2 +
2*q2)*q1*q2*q3^2*q4^2 - 7*(p2^2 + 2*q2)*q1^2*q2*q3^2*q4^2 - 7*(p1^2 +
2*q1)*q1*q2^2*q3^2*q4^2 + 9*q1^2*q2^2*q3^2*q4^2 - ((p4^2 + 2*q4)^2 - 2*q4^2)*(p3^2 +
2*q3)*q1^2*q2^2*q3 - ((p4^2 + 2*q4)^2 - 2*q4^2)*(p2^2 + 2*q2)*q1^2*q2*q3^2 - ((p4^2 +
2*q4)^2 - 2*q4^2)*(p1^2 + 2*q1)*q1*q2^2*q3^2 + ((p4^2 + 2*q4)^2 - 2*q4^2)*q1^2*q2^2*q3^2 ((p3^2 + 2*q3)^2 - 2*q3^2)*(p4^2 + 2*q4)*q1^2*q2^2*q4 - ((p2^2 + 2*q2)^2 - 2*q2^2)*(p4^2 +
2*q4)*q1^2*q3^2*q4 - ((p1^2 + 2*q1)^2 - 2*q1^2)*(p4^2 + 2*q4)*q2^2*q3^2*q4 - ((p3^2 +
2*q3)^2 - 2*q3^2)*(p2^2 + 2*q2)*q1^2*q2*q4^2 - ((p3^2 + 2*q3)^2 - 2*q3^2)*(p1^2 +
2*q1)*q1*q2^2*q4^2 + ((p3^2 + 2*q3)^2 - 2*q3^2)*q1^2*q2^2*q4^2 - ((p2^2 + 2*q2)^2 2*q2^2)*(p3^2 + 2*q3)*q1^2*q3*q4^2 - ((p1^2 + 2*q1)^2 - 2*q1^2)*(p3^2 + 2*q3)*q2^2*q3*q4^2
- ((p2^2 + 2*q2)^2 - 2*q2^2)*(p1^2 + 2*q1)*q1*q3^2*q4^2 + ((p2^2 + 2*q2)^2 2*q2^2)*q1^2*q3^2*q4^2 - ((p1^2 + 2*q1)^2 - 2*q1^2)*(p2^2 + 2*q2)*q2*q3^2*q4^2 + ((p1^2 +
2*q1)^2 - 2*q1^2)*q2^2*q3^2*q4^2)*p1*p2*p3*p4
l 0 1=l ( p1=a1+b1 , p2=a2+b2 , p3=a3+b3 , p4=a4+b4 , q1 =-1* a1 *b1 , q2 =-1* a2 *b2 , q3 =-1*\
a3 *b3 , q4 =-1* a4 *b4 )
b o o l ( l 0 1==g11 )
#This i s a check t o make s u r e t h a t when back s u b s t i t u t e we end up with t h e \
original coefficient .
True
m=l . expand ( )
m
l a t e x (m)
#This i s t h e e x p a n s i o n o f our c o e f f i c i e n t and a l a t e x output f o r e a s y c o p y i n g \
to the t h e s i s .
p1^3*p2^3*p3^3*p4^3*q1*q2*q3*q4 - p1*p2*p3^3*p4^5*q1^2*q2^2*q3 p1*p2^3*p3*p4^5*q1^2*q2*q3^2 - p1^3*p2*p3*p4^5*q1*q2^2*q3^2 - p1*p2*p3^5*p4^3*q1^2*q2^2*q4
- p1*p2^5*p3*p4^3*q1^2*q3^2*q4 - p1^5*p2*p3*p4^3*q2^2*q3^2*q4 p1*p2^3*p3^5*p4*q1^2*q2*q4^2 - p1^3*p2*p3^5*p4*q1*q2^2*q4^2 - p1*p2^5*p3^3*p4*q1^2*q3*q4^2
- p1^5*p2*p3^3*p4*q2^2*q3*q4^2 - p1^3*p2^5*p3*p4*q1*q3^2*q4^2 p1^5*p2^3*p3*p4*q2*q3^2*q4^2 - 5*p1*p2*p3*p4^5*q1^2*q2^2*q3^2 9*p1*p2*p3^3*p4^3*q1^2*q2^2*q3*q4 - 9*p1*p2^3*p3*p4^3*q1^2*q2*q3^2*q4 -

17

9*p1^3*p2*p3*p4^3*q1*q2^2*q3^2*q4 - 5*p1*p2*p3^5*p4*q1^2*q2^2*q4^2 9*p1*p2^3*p3^3*p4*q1^2*q2*q3*q4^2 - 9*p1^3*p2*p3^3*p4*q1*q2^2*q3*q4^2 5*p1*p2^5*p3*p4*q1^2*q3^2*q4^2 - 9*p1^3*p2^3*p3*p4*q1*q2*q3^2*q4^2 5*p1^5*p2*p3*p4*q2^2*q3^2*q4^2 - 31*p1*p2*p3*p4^3*q1^2*q2^2*q3^2*q4 31*p1*p2*p3^3*p4*q1^2*q2^2*q3*q4^2 - 31*p1*p2^3*p3*p4*q1^2*q2*q3^2*q4^2 31*p1^3*p2*p3*p4*q1*q2^2*q3^2*q4^2 - 63*p1*p2*p3*p4*q1^2*q2^2*q3^2*q4^2
p_{1}^{3} p_{2}^{3} p_{3}^{3} p_{4}^{3} q_{1} q_{2} q_{3} q_{4} - p_{1} p_{2} p_{3}^{3}
p_{4}^{5} q_{1}^{2} q_{2}^{2} q_{3} - p_{1} p_{2}^{3} p_{3} p_{4}^{5} q_{1}^{2} q_{2}
q_{3}^{2} - p_{1}^{3} p_{2} p_{3} p_{4}^{5} q_{1} q_{2}^{2} q_{3}^{2} - p_{1} p_{2}
p_{3}^{5} p_{4}^{3} q_{1}^{2} q_{2}^{2} q_{4} - p_{1} p_{2}^{5} p_{3} p_{4}^{3} q_{1}^{2}
q_{3}^{2} q_{4} - p_{1}^{5} p_{2} p_{3} p_{4}^{3} q_{2}^{2} q_{3}^{2} q_{4} - p_{1}
p_{2}^{3} p_{3}^{5} p_{4} q_{1}^{2} q_{2} q_{4}^{2} - p_{1}^{3} p_{2} p_{3}^{5} p_{4}
q_{1} q_{2}^{2} q_{4}^{2} - p_{1} p_{2}^{5} p_{3}^{3} p_{4} q_{1}^{2} q_{3} q_{4}^{2} p_{1}^{5} p_{2} p_{3}^{3} p_{4} q_{2}^{2} q_{3} q_{4}^{2} - p_{1}^{3} p_{2}^{5} p_{3}
p_{4} q_{1} q_{3}^{2} q_{4}^{2} - p_{1}^{5} p_{2}^{3} p_{3} p_{4} q_{2} q_{3}^{2}
q_{4}^{2} - 5 \, p_{1} p_{2} p_{3} p_{4}^{5} q_{1}^{2} q_{2}^{2} q_{3}^{2} - 9 \, p_{1}
p_{2} p_{3}^{3} p_{4}^{3} q_{1}^{2} q_{2}^{2} q_{3} q_{4} - 9 \, p_{1} p_{2}^{3} p_{3}
p_{4}^{3} q_{1}^{2} q_{2} q_{3}^{2} q_{4} - 9 \, p_{1}^{3} p_{2} p_{3} p_{4}^{3} q_{1}
q_{2}^{2} q_{3}^{2} q_{4} - 5 \, p_{1} p_{2} p_{3}^{5} p_{4} q_{1}^{2} q_{2}^{2} q_{4}^{2}
- 9 \, p_{1} p_{2}^{3} p_{3}^{3} p_{4} q_{1}^{2} q_{2} q_{3} q_{4}^{2} - 9 \, p_{1}^{3}
p_{2} p_{3}^{3} p_{4} q_{1} q_{2}^{2} q_{3} q_{4}^{2} - 5 \, p_{1} p_{2}^{5} p_{3} p_{4}
q_{1}^{2} q_{3}^{2} q_{4}^{2} - 9 \, p_{1}^{3} p_{2}^{3} p_{3} p_{4} q_{1} q_{2} q_{3}^{2}
q_{4}^{2} - 5 \, p_{1}^{5} p_{2} p_{3} p_{4} q_{2}^{2} q_{3}^{2} q_{4}^{2} - 31 \, p_{1}
p_{2} p_{3} p_{4}^{3} q_{1}^{2} q_{2}^{2} q_{3}^{2} q_{4} - 31 \, p_{1} p_{2} p_{3}^{3}
p_{4} q_{1}^{2} q_{2}^{2} q_{3} q_{4}^{2} - 31 \, p_{1} p_{2}^{3} p_{3} p_{4} q_{1}^{2}
q_{2} q_{3}^{2} q_{4}^{2} - 31 \, p_{1}^{3} p_{2} p_{3} p_{4} q_{1} q_{2}^{2} q_{3}^{2}
q_{4}^{2} - 63 \, p_{1} p_{2} p_{3} p_{4} q_{1}^{2} q_{2}^{2} q_{3}^{2} q_{4}^{2}

18


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